This article proposes a new denoising method for hyperspectral images (HSIs) corrupted by mixtures (in a statistical sense) of stripe noise, Gaussian noise, and impulsive noise. The proposed method has three distinctive features: 1) it exploits the intrinsic characteristics of HSIs, namely, low-rank and self-similarity; 2) the observation noise is assumed to be additive and modeled by a mixture of Gaussian (MoG) densities; 3) the inference is performed with an expectation maximization (EM) algorithm, which, in addition to the clean HSI, also estimates the mixture parameters (posterior probability of each mode and variances). Comparisons of the proposed method with state-of-the-art algorithms provide experimental evidence of the effectiveness of the proposed denoising algorithm. A MATLAB demo of this work will be available athttps://github.com/TaiXiangJiangfor the sake of reproducibility.

Hyperspectral imaging measures the amount of electromagnetic energy across the instantaneous field of view at a very high resolution in hundreds or thousands of spectral channels. This enables objects to be detected and the identification of materials that have subtle differences between them. However, the increase in spectral resolution often means that there is a decrease in the number of photons received in each channel, which means that the noise linked to the image formation process is greater. This degradation limits the quality of the extracted information and its potential applications. Thus, denoising is a fundamental problem in hyperspectral image (HSI) processing. As images of natural scenes with highly correlated spectral channels, HSIs are characterized by a high level of self-similarity and can be well approximated by low-rank representations. These characteristics underlie the state-of-the-art methods used in HSI denoising. However, where there are rarely occurring pixel types, the denoising performance of these methods is not optimal, and the subsequent detection of these pixels may be compromised. To address these hurdles, in this article, we introduce RhyDe (Robust hyperspectral Denoising), a powerful HSI denoiser, which implements explicit low-rank representation, promotes self-similarity, and, by using a form of collaborative sparsity, preserves rare pixels. The denoising and detection effectiveness of the proposed robust HSI denoiser is illustrated using semireal and real data.

This article proposes a smart hyperspectral sensing strategy, implemented in the spectral domain, conceived for spaceborne sensor systems, where physical space, storage resources, and communication bandwidth are extremely scarce and expensive. Smart sensing means faster and hardware-friendly imaging. Instead of acquiring all band samples in the spectral domain, we randomly select a few band samples per spatial pixel location. A periodic structure of spectral band selector array (SBSA) is designed so that we can learn a subspace basis from subsamples, which is essential to the underlying hyperspectral image (HSI) recovery algorithm. This spectral subsampling sensing strategy yields a demosaicing problem. We propose a blind hyperspectral reconstruction technique termed hyperspectral demosaicing (Hy-demosaicing) exploiting spectral low-rankness and spatial correlation of HSIs. It is blind in the sense that the signal subspace is learned from measured spectral subsamples. The subspace basis is data-adaptive and provides a more compact representation than other non-adaptive representations. This adaptiveness leads to improved image recovery as illustrated in experiments with real data.

This article proposes a denoiser for hyperspectral (HS) images that consider, not only spatial features, but also spectral features. The method starts by projecting the noisy (observed) HS data onto a lower dimensional subspace and then learns a Gaussian mixture model (GMM) from 3-D patches or blocks extracted from the projected data cube. Afterward, the minimum mean squared error (MMSE) estimates of the blocks are obtained in closed form and returned to their original positions. Experiments show that the proposed algorithm is able to outperform other state-of-the-art methods under Gaussian and Poissonian noise and to reconstruct high-quality images in the presence of stripes.

The ever-increasing spectral resolution of hyperspectral images (HSIs) is often obtained at the cost of a decrease in the signal-to-noise ratio of the measurements, thus calling for effective denoising techniques. HSIs from the real world lie in low-dimensional subspaces and are self-similar. The low dimensionality stems from the high correlation existing among the reflectance vectors, and self-similarity is common in real-world images. In this article, we exploit the above two properties. The low dimensionality is a global property that enables the denoising to be formulated just with respect to the subspace representation coefficients, thus greatly improving the denoising performance and reducing the computational complexity during processing. The self-similarity is exploited via a low-rank tensor factorization of nonlocal similar 3-D patches. The proposed factorization hinges on the optimal shrinkage/thresholding of the singular value decomposition (SVD) singular values of low-rank tensor unfoldings. As a result, the proposed method is user friendly and insensitive to its parameters. Its effectiveness is illustrated in a comparison with state-of-the-art competitors. A MATLAB demo of this work is available athttps://github.com/LinaZhuangfor the sake of reproducibility.

Nonnegative blind source separation (nBSS) is often a challenging inverse problem, namely, when the mixing system is ill-conditioned. In this work, we focus on an important nBSS instance, known as hyperspectral unmixing (HU) in remote sensing. HU is a matrix factorization problem aimed at factoring the so-called endmember matrix, holding the material hyperspectral signatures, and the abundance matrix, holding the material fractions at each image pixel. The hyperspectral signatures are usually highly correlated, leading to a fast decay of the singular values (and, hence, high condition number) of the endmember matrix, so HU often introduces an ill-conditioned nBSS scenario. We introduce a new theoretical framework to attack such tough scenarios via the John ellipsoid (JE) in functional analysis. The idea is to identify the maximum volume ellipsoid inscribed in the data convex hull, followed by affinely mapping such ellipsoid into a Euclidean ball. By applying the same affine mapping to the data mixtures, we prove that the endmember matrix associated with the mapped data has condition number 1, the lowest possible, and that these (preconditioned) endmembers form a regular simplex. Exploiting this regular structure, we design a novel nBSS criterion with a provable identifiability guarantee and devise an algorithm to realize the criterion. Moreover, for the first time, the optimization problem for computing JE is exactly solved for a large-scale instance; our solver employs a split augmented Lagrangian shrinkage algorithm with all proximal operators solved by closed-form solutions. The competitiveness of the proposed method is illustrated by numerical simulations and real data experiments.

Combining a high-spatial-resolution multispectral image (HR-MSI) with a low-spatial-resolution hyperspectral image (LR-HSI) has become a common way to enhance the spatial resolution of the HSI. The existing state-of-the-art LR-HSI and HR-MSI fusion methods are mostly based on the matrix factorization, where the matrix data representation may be hard to fully make use of the inherent structures of 3-D HSI. We propose a nonlocal sparse tensor factorization approach, called the NLSTF_SMBF, for the semiblind fusion of HSI and MSI. The proposed method decomposes the HSI into smaller full-band patches (FBPs), which, in turn, are factored as dictionaries of the three HSI modes and a sparse core tensor. This decomposition allows to solve the fusion problem as estimating a sparse core tensor and three dictionaries for each FBP. Similar FBPs are clustered together, and they are assumed to share the same dictionaries to make use of the nonlocal self-similarities of the HSI. For each group, we learn the dictionaries from the observed HR-MSI and LR-HSI. The corresponding sparse core tensor of each FBP is computed via tensor sparse coding. Two distinctive features of NLSTF_SMBF are that: 1) it is blind with respect to the point spread function (PSF) of the hyperspectral sensor and 2) it copes with spatially variant PSFs. The experimental results provide the evidence of the advantages of the NLSTF_SMBF method over the existing state-of-the-art methods, namely, in semiblind scenarios.

Sentinel-2 satellite, launched by the European Space Agency, plays a critical role in various Earth observation missions. However, the spatial resolutions of Sentinel-2 images are different across its spectral bands, including four bands with a resolution of 10 m, six bands with a resolution of 20 m, and three bands with a resolution of 60 m. To facilitate the effectiveness of analyzing these images, super-resolving of the low-/medium-resolution bands to a higher resolution is desired. As in any image restoration inverse problems, we exploit image self-similarity, a commonly observed property in natural images, which underlies the state-of-the-art techniques, e.g., in image denoising. However, the design of self-similarity priors in nondiagonal inverse problems is challenging; often, a denoiser based on self-similarity is plugged into the iterations of an algorithm, without a guarantee of convergence in general. In this article, for the first time, we introduce a convex and scene-adapted regularizer built explicitly on a self-similarity graph directly learned from the Sentinel-2 images. We then develop a fast algorithm, termed Sentinel-2 super-resolution via scene-adapted self-similarity (SSSS). We experimentally show the superiority of SSSS over four commonly observed scenes, indicating the potential usage of our convex self-similarity regularization in other imaging inverse problems.

We investigate the problems of 1-D and 2-D signal recovery from subsampled Hadamard measurements using Haar wavelet as a sparsity inducing prior. These problems are of interest in, e.g., computational imaging applications relying on optical multiplexing or single-pixel imaging. However, the realization of such modalities is often hindered by the coherence between the Hadamard and Haar bases. The variable and multilevel density sampling strategies solve this issue by adjusting the subsampling process to the local and multilevel coherence, respectively, between the two bases; hence enabling successful signal recovery. In this work, we compute an explicit sample-complexity bound for Hadamard-Haar systems as well as uniform and non-uniform recovery guarantees; a seemingly missing result in the related literature. We explore the faithfulness of the numerical simulations to the theoretical results and show in a practically relevant instance, e.g., single-pixel camera, that the target signal can be recovered from a few Hadamard measurements.

Single Pixel (SP) imaging is now a reality in many applications, e.g., biomedical ultrathin endoscope and fluorescent spectroscopy. In this context, many schemes exist to improve the light throughput of these device, e.g., using structured illumination driven by compressive sensing theory. In this work, we consider the combination of SP imaging with Fourier Transform Interferometry (SP-FTI) to reach high-resolution HyperSpectral (HS) imaging, as desirable, e.g., in fluorescent spectroscopy. While this association is not new, we here focus on optimizing the spatial illumination, structured as Hadamard patterns, during the optical path progression. We follow a variable density sampling strategy for space-time coding of the light illumination, and show theoretically and numerically that this scheme allows us to reduce the number of measurements and light-exposure of the observed object compared to conventional compressive SP-FTI.

In satellite imaging remote sensing, injecting spatial details extracted from a panchromatic image into a multispectral image is referred to as pansharpening, which is ill-posed and requires regularization. Self-similarity, a critical prior knowledge yielding great success in regularizing various imaging inverse problems, has been widely observed in natural images; its formalization is not, however, straightforward. Very recently, we mathematically described the self-similarity pattern as a weighted graph, which can then be transformed into an explicit convex regularizer, that is adopted in our pansharpening criterion design. Most importantly, such convexity allows the adoption of convex optimization theory in solving self-similarity regularized inverse problems with convergence guarantee. One step of our pansharpening algorithm is exactly the proximal operator induced by our new self-similarity regularizer, which is solved by another customized algorithm that is interesting in its own right as could be used as a denoiser. Experiments show promising performance of the proposed method.

This paper aims to provide a compact superresolution formulation specific for multispectral (MS) multiresolution optical data, i.e., images characterized by different scales across different spectral bands. The proposed method, named multiresolution sharpening approach (MuSA), relies on the solution of an optimization problem tailored to the properties of those images. The superresolution problem is formulated as the minimization of an objective function containing a data-fitting term that models the blurs and downsamplings of the different bands and a patch-based regularizer that promotes image self-similarity guided by the geometric details provided by the high-resolution bands. By exploiting the approximately low-rank property of the MS data, the ill-posedness of the inverse problem in hand is strongly reduced, thus sharply improving its conditioning. The state-of-the-art color block-matching and 3D filtering (C-BM3D) image denoiser is used as a patch-based regularizer by leveraging the “plug-and-play” framework: the denoiser is plugged into the iterations of the alternating direction method of multipliers. The main novelties of the proposed method are: 1) the introduction of an observation model tailored to the specific properties of (MS) multiresolution images and 2) the exploitation of the high-spatial-resolution bands to guide the grouping step in the color block-matching and 3D filtering (C-BM3D) denoiser, which constitutes a form of regularization learned from the high-resolution channels. The results obtained on the real and synthetic Sentinel 2 data sets give an evidence of the effectiveness of the proposed approach.

Linear hyperspectral unmixing (HU) aims at factoring the observation matrix into an endmember matrix and an abundance matrix. Linear HU via variational minimum volume (MV) regularization has recently received considerable attention in the remote sensing and machine learning areas, mainly owing to its robustness against the absence of pure pixels. We put some popular linear HU formulations under a unifying framework, which involves a data-fitting term and an MV-based regularization term, and collectively solve it via a nonconvex optimization. As the former and the latter terms tend, respectively, to expand (reducing the data-fitting errors) and to shrink the simplex enclosing the measured spectra, it is critical to strike a balance between those two terms. To the best of our knowledge, the existing methods find such balance by tuning a regularization parameter manually, which has little value in unsupervised scenarios. In this paper, we aim at selecting the regularization parameter automatically by exploiting the fact that a too large parameter overshrinks the volume of the simplex defined by the endmembers, making many data points be left outside of the simplex and hence inducing a large data-fitting error, while a sufficiently small parameter yields a large simplex making data-fitting error very small. Roughly speaking, the transition point happens when the simplex still encloses the data cloud but there are data points on all its facets. These observations are systematically formulated to find the transition point that, in turn, yields a good parameter. The competitiveness of the proposed selection criterion is illustrated with simulated and real data.

We propose a new approach to image fusion, inspired by the recent plug-and-play (PnP) framework. In PnP, a denoiser is treated as a black-box and plugged into an iterative algorithm, taking the place of the proximity operator of some convex regularizer, which is formally equivalent to a denoising operation. This approach offers flexibility and excellent performance, but convergence may be hard to analyze, as most state-of-the-art denoisers lack an explicit underlying objective function. Here, we propose using a scene-adapted denoiser (i.e., targeted to the specific scene being imaged) plugged into the iterations of the alternating direction method of multipliers (ADMM). This approach, which is a natural choice for image fusion problems, not only yields state-of-the-art results, but it also allows proving convergence of the resulting algorithm. The proposed method is tested on two different problems: hyperspectral fusion/sharpening and fusion of blurred-noisy image pairs.

Remotely sensed hyperspectral images (HSIs) usually have high spectral resolution but low spatial resolution. A way to increase the spatial resolution of HSIs is to solve a fusion inverse problem, which fuses a low spatial resolution HSI (LR-HSI) with a high spatial resolution multispectral image (HR-MSI) of the same scene. In this paper, we propose a novel HSI super-resolution approach (called LRSR), which formulates the fusion problem as the estimation of a spectral dictionary from the LR-HSI and the respective regression coefficients from both images. The regression coefficients are estimated by formulating a variational regularization problem which promotes local (in the spatial sense) low-rank and sparse regression coefficients. The local regions, where the spectral vectors are low-rank, are estimated by segmenting the HR-MSI. The formulated convex optimization is solved with SALSA. Experiments provide evidence that LRSR is competitive with respect to the state-of-the-art methods.

This paper proposes a new denoising method for hyperspectral images (HSIs) corrupted by mixtures (in a statistical sense) of stripe noise, Gaussian noise, and impulsive noise. The proposed method has three distinctive features: 1) it exploits the intrinsic characteristics of HSIs, namely, low-rank and self-similarity; 2) the observation noise is assumed to be additive and modeled by a mixture of Gaussian (MoG) densities; 3) the inference is performed with an expectation maximization (EM) algorithm, which, in addition to the clean HSI, also estimates the mixture parameters (posterior probability of each mode and variances). Comparisons of the proposed method with state-of-the-art algorithms provide experimental evidence of the effectiveness of the proposed denoising algorithm.

In hyperspectral unmixing and in many other areas (e.g., chemometrics, topic modeling, archetypal analysis) simplex-structured matrix factorization (SSMF) plays an essential role as suggested by years of research efforts devoted to this theme. Specifically, SSMF factorizes a data matrix into two matrix factors with one factor (i.e., the abundances) constrained to have its columns lying in the unit simplex. SSMF criteria include the well-known Craig's seminal minimum-volume enclosing simplex (MVES), originally proposed for blind hyperspectral unmixing, and the recently introduced maximum-volume inscribed ellipsoid (MVIE). The identifiability analysis of those criteria is essential to understand their fundamental behavior and also to devise effective SSMF algorithms tailored to the specificities of the different application scenarios. Our analysis is motivated by a simple fact taking place in most remotely sensed hyperspectral mixtures: in most pixels, only a subset of the materials is present. This is to say that the abundances exhibit a form of sparsity and thus lie in the boundary of the data simplex. We then derive some elegant sufficient condition, showing that as long as data points are locally well spread, perfect SSMF identifiability of both criteria can be guaranteed.

This paper proposes a very light hyperspectral sensing strategy, implemented in the spectral domain, conceived to spaceborne sensor systems, where physical space, storage resources, and communication bandwidth are extremely scarce and expensive. Instead of acquiring all samples in spectral domain, we propose to randomly select a few samples per pixel. This subsampling sensing strategy yields a demosaicing problem. We propose a blind hyperspectral reconstruction technique termed hyperspectral demosaicing (Hy-demosaicing) exploiting low-rank and self-similarity properties of hyperspectral images. It is blind in sense that the signal subspace is learned from measured subsamples. The subspace basis is data adaptive and provides a more compact representation than other non-adaptive representations. This adaptiveness leads to improved image recovery as illustrated in experiments with real data.

Superpixels are a powerful device to characterize the spatial-contextual information in remotely sensed hyperspectral image (HSI) interpretation. However, the exploitation of superpixels in classification problems is not straightforward, often leading to unbearable NP-hard discrete integer optimization problems. In this paper, we attack this hurdle by leveraging on a convex relaxation of the original integer optimization problem, which opens the door to include oversegmented superpixel-based regularizers. Specifically, we develop a new method for generating oversegmented superpixels. Then, we introduce a family of convex regularizers in the form of graph total variation, which promotes the same labeling in each superpixel. Vectorial total variation is also included in order to promote piecewise smoothness and align discontinuities along the class boundaries. The solution of the obtained convex optimization problem is computed with the split-augmented Lagrangian shrinkage algorithm. Experiments on HSIs yield classification maps with precise boundaries and inner consistency inside oversegmented superpixels, leading to the state-of-the-art classification accuracies.

Fusing a low spatial resolution hyperspectral image (LR-HSI) with a high spatial resolution multispectral image (HR-MSI) to obtain a high spatial resolution hyperspectral image (HR-HSI) has attracted increasing interest in recent years. In this paper, we propose a coupled sparse tensor factorization (CSTF) based approach for fusing such images. In the proposed CSTF method, we consider an HR-HSI as a three-dimensional tensor and redefine the fusion problem as the estimation of a core tensor and dictionaries of the three modes. The high spatial-spectral correlations in the HR-HSI are modeled by incorporating a regularizer which promotes sparse core tensors. The estimation of the dictionaries and the core tensor are formulated as a coupled tensor factorization of the LR-HSI and of the HR-MSI. Experiments on two remotely sensed HSIs demonstrate the superiority of the proposed CSTF algorithm over current state-of-the-art HSI-MSI fusion approaches.

This paper presents a novel nonlinear hyperspectral mixture model and its associated supervised unmixing algorithm. The model assumes a linear mixing model corrupted by an additive term which accounts for multiple scattering nonlinearities (NL). The proposed model generalizes bilinear models by taking into account higher order interaction terms. The inference of the abundances and nonlinearity coefficients of this model is formulated as a convex optimization problem suitable for fast estimation algorithms. This formulation accounts for constraints such as the sum-to-one and nonnegativity of the abundances, the non-negativity of the nonlinearity coefficients, and the spatial sparseness of the residuals. The resulting convex problem is solved using the alternating direction method of multipliers (ADMM) whose convergence is ensured theoretically. The proposed mixture model and its unmixing algorithm are validated on both synthetic and real images showing competitive results regarding the quality of the inference and the computational complexity when compared to the state-of-the-art algorithms.

In this paper, we propose a new image denoising method, tailored to specific classes of images, assuming that a dataset of clean images of the same class is available. Similarly to the non-local means (NLM) algorithm, the proposed method computes a weighted average of non-local patches, which we interpret under the importance sampling framework. This viewpoint introduces flexibility regarding the adopted priors, the noise statistics, and the computation of Bayesian estimates. The importance sampling viewpoint is exploited to approximate the minimum mean squared error (MMSE) patch estimates, using the true underlying prior on image patches. The estimates thus obtained converge to the true MMSE estimates, as the number of samples approaches infinity. Experimental results provide evidence that the proposed denoiser outperforms the state-of-the-art in the specific classes of face and text images.

In this paper, we address the problem of recovering images degraded by Poisson noise, where the image is known to belong to a specific class. In the proposed method, a dataset of clean patches from images of the class of interest is clustered using multivariate Gaussian distributions. In order to recover the noisy image, each noisy patch is assigned to one of these distributions, and the corresponding minimum mean squared error (MMSE) estimate is obtained. We propose to use a self-normalized importance sampling approach, which is a method of the Monte-Carlo family, for the both determining the most likely distribution and approximating the MMSE estimate of the clean patch. Experimental results shows that our proposed method outperforms other methods for Poisson denoising at a low SNR regime.

The ever increasing spectral resolution of the hyperspectral images (HSIs) is often obtained at the cost of a decrease in the signal-to-noise of the measurements, thus calling for effective denoising techniques. HSIs from the real world live in low dimensional subspaces and are self-similar. The low dimensionality stems from the high correlation existing among the reflectance vectors and the self-similarity is common to images of the real world. In this paper, we exploit the above two properties. The low dimensionality is a global property, which enables the denoising to be formulated just with respect to the subspace representation coefficients, thus greatly improving the denoising performance and reducing the processing computational complexity. The self-similarity is exploited via low-rank tensor factorization of non-local similar 3D-patches. The proposed factorization hinges on optimal shrinkage/thresholding of SVD singular value of low-rank tensor unfoldings. As a result, the proposed method has no parameters, apart from the noise variance. Its effectiveness is illustrated in a comparison with state-of-the-art competitors.

The synergistic analysis of light detection and ranging (LiDAR) and hyperspectral data is attracting a significant interest in recent years due to the complementary nature of these two sources of remote sensing data. In this paper, we propose a new spectral-spatial classification method able to jointly exploit these two kinds of data. Our work is based on three innovative components: 1) a superpixel generation method aimed at multivariate image spatial partitioning, 2) a multi-source framework for feature extraction, and 3) a convex framework used to approach the solutions of the resulted image labeling problem associated with vectorial total variation and superpixel-based graph total variation regularizers. Our experimental results, conducted with a hyperspectral data set collected by the Compact Airborne Spectrographic Imager (CASI) spectrometer over the city of Houston in 2013 and a corresponding LiDAR data set, illustrate the effectiveness of the proposed framework.

This work introduces a cloud shadow removal method for hyperspectral images (HSIs) based on hyperspectral unmixing. The shading is modeled by a spectral offset and a spectral-dependent attenuation. The offset and the attenuation are estimated by solving a nonconvex optimization problem, which exploits the linear mixing model (LMM). The mixing matrix of the LMM is estimated from the unshadowed image areas. The effectiveness of the proposed method is assessed from classification results of the Houston 2013 (Compact Airborne Spectrographic Imager (CASI) spectrometer VHR HS), whose shadowed areas were removed with the proposed method. The obtained results indicate classification performances in the shadowed areas very close to those of the unshadowed ones, thus providing evidence of the effectiveness of the proposed shading removal technique.

Over the last decade, the number of missions including hyperspectral cameras of increasing resolution has grown considerably. Traditional compression techniques have been proposed as an efficient way to store and transmit the ever increasing amount of hyperspectral data and to cope with the limited transmission bandwidth between the onboard systems and the ground stations. On the other hand, compressive sensing techniques try to deal with the same issue from a different perspective: the signal is compressed while the acquisition takes place, and the decompression is carried out by solving an optimization problem. Due to scarce onboard resources, the compressing sensing paradigm in hyperspectral systems is attractive, namely because the main computational burden to recover the original data is carried out in the ground stations, where more computational resources are available. In this paper, we develop a new technique for compressive sensing of hyperspectral images, where the measurement process takes place both in the spatial and the spectral domains by efficiently exploiting the high correlation of hyperspectral images. The proposed technique was tested using synthetic and semi-real images yielding competitive results.

In this paper, we focus the attention on the superresolution of multispectral (MS) multiresolution images (e.g., Sentinel 2, Aster, MODIS). By taking advantage of the high spatial resolution bands, we minimize an objective function containing a quadratic data fitting term, an edge preserving regularizer, and a patch-based plug-and play prior promoting self-similar images. To cope with the ill-posedness of the problem we i) exploit the fact that the images are approximately low-rank, and ii) propose a hierarchical method which sharpens in the first place the medium resolution bands and then the coarse resolution ones. The optimization is solved with the alternating direction method of multipliers (ADMM), yielding a fast, flexible, and effective solver, named Superresolution MUltiband multireSolution Hierarchical approach (SMUSH). Quantitative and qualitative results obtained on simulated and real Sentinel 2 (S2) images show the SMUSH effectiveness.

Abundance information has been recently used to assist hyperspectral image classification by combining the information coming from classification and unmixing. The fact that classes are usually inconsistent with endmembers makes it a crucial issue to find possible connections between classification and unmixing. This paper describes a new class-based endmember extraction and sparse unmixing approach aimed at establishing the correspondence between endmembers and classes. The proposed approach is exploited in a semisupervised classification framework that combines classification and unmixing with active learning (AL). During the AL process, the class probabilities and abundance information are exploited simultaneously to select the most informative unlabeled samples for classification purposes. Our approach adopts a well-established discriminative probabilistic classifier, the multinomial logistic regression (MLR), to learn the class posterior probabilities. The effectiveness of the proposed method is evaluated using real hyperspectral data set collected by the NASA Jet Propulsion Laboratory's Airborne Visible Infrared Imaging Spectrometer (AVIRIS) over the Indian Pines region, Indiana.

Hyperspectral images (HSIs) cover hundreds of narrow spectral bands, thus yielding high spectral resolution, enabling precise identification of different materials. However, the existence of dead pixels in the light sensors produces a number of irrelevant measurements, which may compromise the usefulness of HSIs. In this paper, a new hyperspectral inpainting method, named HyInpaint, is proposed. The original HSI is represented on a low dimensional subspace and its estimation is formalized with respect to the subspace representation coefficients on a given basis. The coefficients are estimated by minimizing an objective function which, in addition to the data term, contains a regularizer based on the Criminisi's inpainting method. The optimization is carried out by an instance of the alternating direction method of multipliers (ADMM), adopting the plug-and-play methodology. The effectiveness of the proposed HyInpaint approach is illustrated on Tiangong-1 hyperspectral visible near infrared (VNIR) wavebands data.

Point cloud rendering has a vital role in the user Quality of Experience for applications adopting point cloud based representations. While this is not a new area, it has recently become more relevant with the recent interest on point cloud coding by major standardization groups, notably JPEG and MPEG. The screened Poisson surface reconstruction is a state-of-the-art technique for generating a watertight surface mesh from the point cloud samples. While its screening component allows the surface to better fit the cloud points, this fitting may lead to undesired artifacts in the surface, notably when the point cloud is noisy. This paper proposes to improve this reconstruction method by making it more robust to noise by adopting a generalized Tikhonov regularization term. The proposed regularization approach smooths regions that should be flat while keeping the important details in the edges, thus creating more pleasant surface reconstructions.

Blind hyperspectral unmixing jointly estimates spectral signatures and abundances in hyperspectral images (HSIs). Hyperspectral unmixing is a powerful tool for analyzing hyperspectral data. However, the usual huge size of HSIs may raise difficulties for classical unmixing algorithms, namely, due to limitations of the hardware used. Therefore, some researchers have considered distributed algorithms. In this paper, we develop a distributed hyperspectral unmixing algorithm that uses the alternating direction method of multipliers and ℓ1sparse regularization. The hyperspectral unmixing problem is split into a number of smaller subproblems that are individually solved, and then the solutions are combined. A key feature of the proposed algorithm is that each subproblem does not need to have access to the whole HSI. The algorithm may also be applied to multitemporal HSIs with due adaptations accounting for variability that often appears in multitemporal images. The effectiveness of the proposed algorithm is evaluated using both simulated data and real HSIs.

This paper proposes using a Gaussian mixture model as a patch-based prior, for solving two image inverse problems, namely image deblurring and compressive imaging. We capitalize on the fact that variable splitting algorithms, like ADMM, are able to decouple the handling of the observation operator from that of the regularizer, and plug a state-of-the-art algorithm into the denoising step. Furthermore, we show that, when applied to a specific type of image, a Gaussian mixture model trained from an database of images of the same type is able to outperform current state-of-the-art generic methods.

Sparse hyperspectral unmixing is a widely used technique in remote sensing data characterization. It aims at inferring, from a large spectral library, the pure spectral signatures (endmembers) present in each pixel of a hyperspectral image, jointly with their corresponding abundances. The input to sparse unmixing is represented, thus, by a hyperspectral image acquired from a platform flying at high altitude and a spectral library compiled using laboratory measurements. The reflectance datacube results from a complex ensemble of algorithms which translate the digital numbers stored by the sensor to meaningful ground reflectance, including the removal of atmospheric influence. A recurrent question in the research community does not have an answer yet: how does the atmospheric composition at the time of the flight influence the fractional abundances retrieved via sparse unmixing? This is a fundamental question, as the atmospheric parameters are subject to uncertainties, being very difficult to know them in all pixels. In this paper, we investigate how the uncertainty in two atmospheric parameters: water vapor content and visibility range, propagates to the final abundance maps via atmospheric correction of the sensed image. Our experiments reveal that sparse unmixing is more robust to uncertainty in those parameters and performs better in terms of accuracy than unmixing with image-based endmembers.

The superpixels provided by an unsupervised segmentation algorithm are sets of neighboring pixels homogeneous in some sense. Therefore it is very likely that, in a classification problem, most pixels in a superpixel belong to the same class, namely if the homogeneity criterion is compatible with the class statistics. Superpixels are, therefore, a powerful device to express spatial contextual information. However, the exploitation of superpixels in a principled way is not straightforward. Recent efforts attack this problem under a discrete optimization framework, by including regularization terms promoting consistence of the labels in the superpixels and computing approximate labelings with graph-cut algorithms. The well known hardness of integer optimization problems is a major limitation of this line of attack. In this paper, we introduce a new strategy, based on convex relaxation, to include the spatial information provided by superpixels in classification problems. The convex relaxation of an integer optimization problem opens a door to include extra information, such as spatial partitioning information given by over-segmented superpixels. The convex optimization problem thus obtained is solved by using SALSA algorithm. Experimental results with the ROSIS Pavia University dataset illustrate the effectiveness of the proposed framework.

Hyperspectral data compression and dimensionality reduction has received considerable interest in recent years due to the high spectral resolution of these images. Contrarily to the conventional dimensionality reduction schemes, the spectral compressive acquisition method (SpeCA) performs dimensionality reduction based on random projections. The SpeCA methodology has applications in Hyperspectral Compressive Sensing and also in dimensionality reduction. Due to the extremely large volumes of data collected by imaging spectrometers, high performance computing architectures are needed for data compression of high dimensional hyperspectral data under real-time constrained applications. In this paper a parallel implementation of SpeCA on Graphics Processing Units (GPUs) using the compute unified device architecture (CUDA) is proposed. The proposed implementation is performed in a pixel-by-pixel fashion using coalesced accesses to memory and exploiting shared memory to store temporary data. Furthermore, the kernels have been optimized to minimize the threads divergence, therefore, achieving high GPU occupancy. The experimental results obtained for simulated and real hyperspectral data sets reveal speedups up to 21 times, which demonstrates that the GPU implementation can significantly accelerate the methods execution over big datasets while maintaining the methods accuracy.

Blind hyperspectral unmixing is the task of jointly estimating the spectral signatures of material in a hyperspectral images and their abundances at each pixel. The size of hyperspectral images are usually very large, which may raise difficulties for classical optimization algorithms, due to limited memory of the hardware used. One solution to this problem is to consider distributed algorithms. In this paper, we develop a distributed sparse hyperspectral unmixing algorithm using the alternating direction method of multipliers (ADMM) algorithm and ℓ1sparse regularization. Each sub-problem does not need to have access to the whole hyperspectral image. The algorithm is evaluated using a very large real hyperspectral image.

The very high spectral resolution of Hyperspectral Images (HSIs) enables the identification of materials with subtle differences and the extraction subpixel information. However, the increasing of spectral resolution often implies an increasing in the noise linked with the image formation process. This degradation mechanism limits the quality of extracted information and its potential applications. This paper presents a new HSI denoising approach developed under the assumption that the clean HSI is low-rank and self-similar. Under these assumptions, the clean HSI admits extremely compact and sparse representations, which are exploited to derive a very fast and competitive denoising algorithm, named Fast Hyperspectral Denoising (FastHyDe), able to cope with Gaussian and Poissonian noise. In a series of experiments, the proposed approach competes with state-of-the-art methods, with much lower computational complexity.

This letter proposes a robust fast multiband image fusion method to merge a high-spatial low-spectral resolution image and a low-spatial high-spectral resolution image. Following the method recently developed by Wei et al., the generalized Sylvester matrix equation associated with the multiband image fusion problem is solved in a more robust and efficient way by exploiting the Woodbury formula, avoiding any permutation operation in the frequency domain as well as the blurring kernel invertibility assumption required in their method. Thanks to this improvement, the proposed algorithm requires fewer computational operations and is also more robust with respect to the blurring kernel compared with the one developed by Wei et al. The proposed new algorithm is tested with different priors considered by Wei et al. Our conclusion is that the proposed fusion algorithm is more robust than the one by Wei et al. with a reduced computational cost.

The dictionary-aided sparse regression (SR) approach has recently emerged as a promising alternative to hyperspectral unmixing (HU) in remote sensing. By using an available spectral library as a dictionary, the SR approach identifies the underlying materials in a given hyperspectral image by selecting a small subset of spectral samples in the dictionary to represent the whole image. A drawback with the current SR developments is that an actual spectral signature in the scene is often assumed to have zero mismatch with its corresponding dictionary sample, and such an assumption is considered too ideal in practice. In this paper, we tackle the spectral signature mismatch problem by proposing a dictionary-adjusted nonconvex sparsity-encouraging regression (DANSER) framework. The main idea is to incorporate dictionary correcting variables in an SR formulation. A simple and low per-iteration complexity algorithm is tailor-designed for practical realization of DANSER. Using the same dictionary correcting idea, we also propose a robust subspace solution for dictionary pruning. Extensive simulations and real-data experiments show that the proposed method is effective in mitigating the undesirable spectral signature mismatch effects.

Spectral unmixing is an important technique for remotely sensed hyperspectral data exploitation. It amounts to identifying a set of pure spectral signatures, which are called endmembers, and their corresponding fractional, draftrulesabun-dances in each pixel of the hyperspectral image. Over the last years, different algorithms have been developed for each of the three main steps of the spectral unmixing chain: 1) estimation of the number of endmembers in a scene; 2) identification of the spectral signatures of the endmembers; and 3) estimation of the fractional abundance of each endmember in each pixel of the scene. However, few algorithms can perform all the stages involved in the hyperspectral unmixing process. Such algorithms are highly desirable to avoid the propagation of errors within the chain. In this paper, we develop a new algorithm, which is termed robust collaborative nonnegative matrix factorization (R-CoNMF), that can perform the three steps of the hyperspectral unmixing chain. In comparison with other conventional methods, R-CoNMF starts with an overestimated number of endmembers and removes the redundant endmembers by means of collaborative regularization. Our experimental results indicate that the proposed method provides better or competitive performance when compared with other widely used methods.

This paper presents a multiband image fusion algorithm based on unsupervised spectral unmixing for combining a high-spatial-low-spectral-resolution image and a low-spatial-high-spectral-resolution image. The widely used linear observation model (with additive Gaussian noise) is combined with the linear spectral mixture model to form the likelihoods of the observations. The nonnegativity and sum-to-one constraints resulting from the intrinsic physical properties of the abundances are introduced as prior information to regularize this ill-posed problem. The joint fusion and unmixing problem is then formulated as maximizing the joint posterior distribution with respect to the endmember signatures and abundance maps. This optimization problem is attacked with an alternating optimization strategy. The two resulting subproblems are convex and are solved efficiently using the alternating direction method of multipliers. Experiments are conducted for both synthetic and semi-real data. Simulation results show that the proposed unmixing-based fusion scheme improves both the abundance and endmember estimation compared with the state-of-the-art joint fusion and unmixing algorithms.

This paper presents three hyperspectral mixture models jointly with Bayesian algorithms for supervised hyperspectral unmixing. Based on the residual component analysis model, the proposed general formulation assumes the linear model to be corrupted by an additive term whose expression can be adapted to account for nonlinearities (NLs), endmember variability (EV), or mismodeling effects (MEs). The NL effect is introduced by considering a polynomial expression that is related to bilinear models. The proposed new formulation of EV accounts for shape and scale endmember changes while enforcing a smooth spectral/spatial variation. The ME formulation considers the effect of outliers and copes with some types of EV and NL. The known constraints on the parameter of each observation model are modeled via suitable priors. The posterior distribution associated with each Bayesian model is optimized using a coordinate descent algorithm, which allows the computation of the maximum a posteriori estimator of the unknown model parameters. The proposed mixture and Bayesian models and their estimation algorithms are validated on both synthetic and real images showing competitive results regarding the quality of the inferences and the computational complexity, when compared with the state-of-the-art algorithms.

In image deconvolution problems, the diagonalization of the underlying operators by means of the fast Fourier transform (FFT) usually yields very large speedups. When there are incomplete observations (e.g., in the case of unknown boundaries), standard deconvolution techniques normally involve non-diagonalizable operators, resulting in rather slow methods or, otherwise, use inexact convolution models, resulting in the occurrence of artifacts in the enhanced images. In this paper, we propose a new deconvolution framework for images with incomplete observations that allows us to work with diagonalized convolution operators, and therefore is very fast. We iteratively alternate the estimation of the unknown pixels and of the deconvolved image, using, e.g., an FFT-based deconvolution method. This framework is an efficient, high-quality alternative to existing methods of dealing with the image boundaries, such as edge tapering. It can be used with any fast deconvolution method. We give an example in which a state-of-the-art method that assumes periodic boundary conditions is extended, using this framework, to unknown boundary conditions. Furthermore, we propose a specific implementation of this framework, based on the alternating direction method of multipliers (ADMM). We provide a proof of convergence for the resulting algorithm, which can be seen as a "partial" ADMM, in which not all variables are dualized. We report experimental comparisons with other primal-dual methods, where the proposed one performed at the level of the state of the art. Four different kinds of applications were tested in the experiments: deconvolution, deconvolution with inpainting, superresolution, and demosaicing, all with unknown boundaries.

Remote sensing hyperspectral images (HSIs) are quite often low rank, in the sense that the data belong to a low dimensional subspace/manifold. This has been recently exploited for the fusion of low spatial resolution HSI with high spatial resolution multispectral images in order to obtain super-resolution HSI. Most approaches adopt an unmixing or a matrix factorization perspective. The derived methods have led to state-of-the-art results when the spectral information lies in a low-dimensional subspace/manifold. However, if the subspace/manifold dimensionality spanned by the complete data set is large, i.e., larger than the number of multispectral bands, the performance of these methods mainly decreases because the underlying sparse regression problem is severely ill-posed. In this paper, we propose a local approach to cope with this difficulty. Fundamentally, we exploit the fact that real world HSIs are locally low rank, that is, pixels acquired from a given spatial neighborhood span a very low-dimensional subspace/manifold, i.e., lower or equal than the number of multispectral bands. Thus, we propose to partition the image into patches and solve the data fusion problem independently for each patch. This way, in each patch the subspace/manifold dimensionality is low enough, such that the problem is not ill-posed anymore. We propose two alternative approaches to define the hyperspectral super-resolution through local dictionary learning using endmember induction algorithms. We also explore two alternatives to define the local regions, using sliding windows and binary partition trees. The effectiveness of the proposed approaches is illustrated with synthetic and semi real data.

This paper presents a new parallel implementation of a previously hyperspectral coded aperture (HYCA) algorithm for compressive sensing on graphics processing units (GPUs). HYCA method combines the ideas of spectral unmixing and compressive sensing exploiting the high spatial correlation that can be observed in the data and the generally low number of endmembers needed in order to explain the data. The proposed implementation exploits the GPU architecture at low level, thus taking full advantage of the computational power of GPUs using shared memory and coalesced accesses to memory. The proposed algorithm is evaluated not only in terms of reconstruction error but also in terms of computational performance using two different GPU architectures by NVIDIA: GeForce GTX 590 and GeForce GTX TITAN. Experimental results using real data reveals signficant speedups up with regards to serial implementation.

Hyperspectral image classification is a challenging classification problem: obtaining complete and representative training sets is costly; pixels can belong to unknown classes; and it is generally an ill-posed problem. The need to achieve high classification accuracy surpasses the need to classify the entire image. To achieve this, we use classification with rejection by providing the classifier an option not to classify a pixel and consequently reject it. We propose a method for supervised hyperspectral image classification combining the use of contextual priors with classification with rejection. Rejection is introduced as an extra class that models the probability of classifier failure. We validate the resulting algorithm in the AVIRIS Indian Pines scene and illustrate the performance increase resulting from classification with rejection.

Hyperspectral data compression has received considerable interest in recent years. Contrarily to the conventional compression schemes, which first acquire the full data set and then implement some compressing technique, compressive sensing (CS) acquires directly the compressed signal which will be later recovered on the ground station. The CS paradigm fits perfectly the requirements of onborad hyperspectral imaging systems in terms of energy, computing power, and bandwidth. By using CS in these systems, the amount of data acquired and transmitted to the ground stations is reduced and the bulk of the computation to infer the original data is carried out in the ground stations. In this paper, we present a new technique to perform CS of hyperspectral images (HSIs), which exploit the fact that HSIs admits a low dimensional linear representation. The proposed method is blind in the sense that linear representation is learned with low computational cost from the compressed measurements. Furthermore the proposed method is very light from the computational point of view and it can recover perfectly the original image in noise-free scenarios. The effectiveness of the proposed method is illustrated in both synthetic and real scenarios.

Compressive Sensing has raised as a very useful way to save costs in the acquisition equipment due to the fact that with this technique we can measure the signal in an already compressed form. This is very interesting in hyperspectral applications due to the large amount of data that the hyper-spectral sensors collect, store and transmit to the ground stations. Over the last years many compressive sensing methods have been applied to hyperspectral images, and others have been proposed for exploiting the unique features of this kind of images. Over the last years, many techniques have been proposed to perform compressive sensing in hyperspectral imaging. One of them is the Hyperspectral Coded Aperture (HYCA), which exploits two characteristics of hyper-spectral imagery: 1) the hyperspectral vectors belong to a low dimensional subspace, and 2) the data cube components exhibit very high correlation in the spatial and in the spectral domains. However, HYCA requires the knowledge of the subspace in advance, which, very often, may compromise its applicability. In this paper it is presented a new technique similar to HYCA which does not require the knowledge of the subspace in advance; the proposed technique is termed blind HYCA (B-HYCA) and it performs a form of blind hyperspectral compressed sensing.

This paper addresses interferometric phase image estimation, i.e., the estimation of phase modulo-2π images from sinusoidal 2π-periodic and noisy observations. These degradation mechanisms make interferometric phase image estimation a quite challenging problem. We tackle this challenge by reformulating the true estimation problem as a sparse regression, often termed sparse coding, in the complex domain. Following the standard procedure in patch-based image restoration, the image is partitioned into small overlapping square patches, and the vector corresponding to each patch is modeled as a sparse linear combination of vectors, termed the atoms, taken from a set called dictionary. Aiming at optimal sparse representations, and thus at optimal noise removing capabilities, the dictionary is learned from the data that it represents via matrix factorization with sparsity constraints on the code (i.e., the regression coefficients) enforced by the ℓ1norm. The effectiveness of the new sparse-coding-based approach to interferometric phase estimation, termed the SpInPHASE, is illustrated in a series of experiments with simulated and real data where it outperforms the state-of-the-art.

Linear spectral unmixing aims at estimating the number of pure spectral substances, also calledendmembers, their spectral signatures, and their abundance fractions in remotely sensed hyperspectral images. This paper describes a method for unsupervised hyperspectral unmixing called minimum volume simplex analysis (MVSA) and introduces a new computationally efficient implementation. MVSA approaches hyperspectral unmixing by fitting a minimum volume simplex to the hyperspectral data, constraining the abundance fractions to belong to the probability simplex. The resulting optimization problem, which is computationally complex, is solved in this paper by implementing a sequence of quadratically constrained subproblems using the interior point method, which is particularly effective from the computational viewpoint. The proposed implementation (available online: www.lx.it.pt/%7ejun/DemoMVSA.zip) is shown to exhibit state-of-the-art performance not only in terms of unmixing accuracy, particularly in nonpure pixel scenarios, but also in terms of computational performance. Our experiments have been conducted using both synthetic and real data sets. An important assumption of MVSA is that pure pixels may not be present in the hyperspectral data, thus addressing a common situation in real scenarios which are often dominated by highly mixed pixels. In our experiments, we observe that MVSA yields competitive performance when compared with other available algorithms that work under the nonpure pixel regime. Our results also demonstrate that MVSA is well suited to problems involving a high number of endmembers (i.e., complex scenes) and also for problems involving a high number of pixels (i.e., large scenes).

Abstract—Hyperspectral image classification has been an active topic of research in recent years. In the past, many different types of features have been extracted (using both linear and nonlinear strategies) for classification problems. On the one hand, some approaches have exploited the original spectral information or other features linearly derived from such information in order to have classes which are linearly separable. On the other hand, other techniques have exploited features obtained through nonlinear transformations intended to reduce data dimensionality, to better model the inherent nonlinearity of the original data (e.g., kernels) or to adequately exploit the spatial information contained in the scene (e.g., using morphological analysis). Special attention has been given to techniques able to exploit a single kind of features, such as composite kernel learning or multiple kernel learning, developed in order to deal with multiple kernels. However, few

Hyperspectral imaging relies on sophisticated acquisition and data processing systems able to acquire, process, store, and transmit hundreds or thousands of image bands from a given area of interest. In this paper, we exploit the high correlation existing among the components of the hyperspectral data sets to introduce a new compressive sensing methodology, termed hyperspectral coded aperture (HYCA), which largely reduces the number of measurements necessary to correctly reconstruct the original data. HYCA relies on two central properties of most hyperspectral images, usually termed data cubes: 1) the spectral vectors live on a low-dimensional subspace; and 2) the spectral bands present high correlation in both the spatial and the spectral domain. The former property allows to represent the data vectors using a small number of coordinates. In this paper, we particularly exploit the high spatial correlation mentioned in the latter property, which implies that each coordinate is piecewise smooth and thus compressible using local differences. The measurement matrix computes a small number of random projections for every spectral vector, which is connected with coded aperture schemes. The reconstruction of the data cube is obtained by solving a convex optimization problem containing a data term linked to the measurement matrix and a total variation regularizer. The solution of this optimization problem is obtained by an instance of the alternating direction method of multipliers that decomposes very hard problems into a cyclic sequence of simpler problems. In order to address the need to set up the parameters involved in the HYCA algorithm, we also develop a constrained version of HYCA (C-HYCA), in which all the parameters can be automatically estimated, which is an important aspect for practical application of the algorithm. A series of experiments with simulated and real data shows the effectiveness of HYCA and C-HYCA, indicating their potential in real-world applications.

Hyperspectral remote sensing images (HSIs) usually have high spectral resolution and low spatial resolution. Conversely, multispectral images (MSIs) usually have low spectral and high spatial resolutions. The problem of inferring images that combine the high spectral and high spatial resolutions of HSIs and MSIs, respectively, is a data fusion problem that has been the focus of recent active research due to the increasing availability of HSIs and MSIs retrieved from the same geographical area. We formulate this problem as the minimization of a convex objective function containing two quadratic data-fitting terms and an edge-preserving regularizer. The data-fitting terms account for blur, different resolutions, and additive noise. The regularizer, a form of vector total variation, promotes piecewise-smooth solutions with discontinuities aligned across the hyperspectral bands. The downsampling operator accounting for the different spatial resolutions, the nonquadratic and nonsmooth nature of the regularizer, and the very large size of the HSI to be estimated lead to a hard optimization problem. We deal with these difficulties by exploiting the fact that HSIs generally “live” in a low-dimensional subspace and by tailoring the split augmented Lagrangian shrinkage algorithm (SALSA), which is an instance of the alternating direction method of multipliers (ADMM), to this optimization problem, by means of a convenient variable splitting. The spatial blur and the spectral linear operators linked, respectively, with the HSI and MSI acquisition processes are also estimated, and we obtain an effective algorithm that outperforms the state of the art, as illustrated in a series of experiments with simulated and real-life data.

Many powerful pansharpening approaches exploit the functional relation between the fusion of PANchromatic (PAN) and MultiSpectral (MS) images. To this purpose, the modulation transfer function of the MS sensor is typically used, being easily approximated as a Gaussian filter whose analytic expression is fully specified by the sensor gain at the Nyquist frequency. However, this characterization is often inadequate in practice. In this paper, we develop an algorithm for estimating the relation between PAN and MS images directly from the available data through an efficient optimization procedure. The effectiveness of the approach is validated both on a reduced scale data set generated by degrading images acquired by the IKONOS sensor and on full-scale data consisting of images collected by the QuickBird sensor. In the first case, the proposed method achieves performances very similar to that of the algorithm that relies upon the full knowledge of the degrading filter. In the second, it is shown to outperform several very credited state-of-the-art approaches for the extraction of the details used in the current literature.

This paper presents a variational-based approach for fusing hyperspectral and multispectral images. The fusion problem is formulated as an inverse problem whose solution is the target image assumed to live in a lower dimensional subspace. A sparse regularization term is carefully designed, relying on a decomposition of the scene on a set of dictionaries. The dictionary atoms and the supports of the corresponding active coding coefficients are learned from the observed images. Then, conditionally on these dictionaries and supports, the fusion problem is solved via alternating optimization with respect to the target image (using the alternating direction method of multipliers) and the coding coefficients. Simulation results demonstrate the efficiency of the proposed algorithm when compared with state-of-the-art fusion methods.

This paper presents a new Bayesian collaborative sparse regression method for linear unmixing of hyperspectral images. Our contribution is twofold; first, we propose a new Bayesian model for structured sparse regression in which the supports of the sparse abundance vectors are a priori spatially correlated across pixels (i.e., materials are spatially organized rather than randomly distributed at a pixel level). This prior information is encoded in the model through a truncated multivariate Ising Markov random field, which also takes into consideration the facts that pixels cannot be empty (i.e., there is at least one material present in each pixel), and that different materials may exhibit different degrees of spatial regularity. Second, we propose an advanced Markov chain Monte Carlo algorithm to estimate the posterior probabilities that materials are present or absent in each pixel, and, conditionally to the maximum marginal a posteriori configuration of the support, compute the minimum mean squared error estimates of the abundance vectors. A remarkable property of this algorithm is that it self-adjusts the values of the parameters of the Markov random field, thus relieving practitioners from setting regularization parameters by cross-validation. The performance of the proposed methodology is finally demonstrated through a series of experiments with synthetic and real data and comparisons with other algorithms from the literature.

The paper addresses interferometric phase image estimation, that is, the estimation of phase modulo-2π images from sinusoidal 2π-periodic and noisy observations. These degradation mechanisms make interferometric phase image estimation a challenging problem. We tackle this challenge by reformulating the true estimation problem as a sparse regression in the complex domain. Following the standard procedure in patch-based image restoration, the image is partitioned into small overlapping square patches. BM3D algorithm equipped with high order SVD (HOSVD) is used to form complex domain frames suitable to sparse representations of the complex-valued data. HOSVD applied to the groups of BM3D data enables the design of spatially variant and data adaptive orthonormal complex domain transforms. The effectiveness of the new sparse coding based approach to interferometric phase estimation, termed Interferometric PHASE via Block matching and High order SVD (InPHASE-BHS) is illustrated in a series of simulation experiments where it outperforms the state-of-the-art.

Hyperspectral remote sensing images (HSIs) are characterized by having a low spatial resolution and a high spectral resolution, whereas multispectral images (MSIs) are characterized by low spectral and high spatial resolutions. These complementary characteristics have stimulated active research in the inference of images with high spatial and spectral resolutions from HSI-MSI pairs. In this paper, we formulate this data fusion problem as the minimization of a convex objective function containing two data-fitting terms and an edge-preserving regularizer. The data-fitting terms are quadratic and account for blur, different spatial resolutions, and additive noise; the regularizer, a form of vector Total Variation, promotes aligned discontinuities across the reconstructed hyperspectral bands. The optimization described above is rather hard, owing to its non-diagonalizable linear operators, to the non-quadratic and non-smooth nature of the regularizer, and to the very large size of the image to be inferred. We tackle these difficulties by tailoring the Split Augmented Lagrangian Shrinkage Algorithm (SALSA) - an instance of the Alternating Direction Method of Multipliers (ADMM) - to this optimization problem. By using a convenient variable splitting and by exploiting the fact that HSIs generally “live” in a low-dimensional subspace, we obtain an effective algorithm that yields state-of-the-art results, as illustrated by experiments.

Remote sensing hyperspectral images (HSI) are quite often locally low rank, in the sense that the spectral vectors acquired from a given spatial neighborhood belong to a low dimensional subspace/manifold. This has been recently exploited for the fusion of low spatial resolution HSI with high spatial resolution multispectral images (MSI) in order to obtain super-resolution HSI. Most approaches adopt an unmixing or a matrix factorization perspective. The derived methods have led to state-of-the-art results when the spectral information lies in a low dimensional subspace/manifold. However, if the subspace/manifold dimensionality spanned by the complete data set is large, the performance of these methods decrease mainly because the underlying sparse regression is severely ill-posed. In this paper, we propose a local approach to cope with this difficulty. Fundamentally, we exploit the fact that real world HSI are locally low rank, to partition the image into patches and solve the data fusion problem independently for each patch. This way, in each patch the subspace/manifold dimensionality is low enough to obtain useful super-resolution. We explore two alternatives to define the local regions, using sliding windows and binary partition trees. The effectiveness of the proposed approach is illustrated with synthetic and semi-real data.

This paper presents a new multiple feature learning approach for accurate spectral-spatial classification of hyperspec-tral images. The proposed method integrates multiple features based on the logarithmic opinion pool. We consider subspace multinomial logistic regression for classification as it exhibits a flexible structure for the combination of multiple features through the posterior probability. At the same time, it is able to cope with highly mixed hyperspectral data and with the presence of limited training samples. In this work, we considered lowpass filtering and morphological attribute profiles for spatial feature extraction. Our experimental results with a real hyperspectral images collected by the NASA Jet Propulsion Laboratory's Airborne Visible Infra-Red Imaging Spectrometer (AVIRIS) indicate that the proposed method exhibits state-of-the-art classification performance.

In this paper, we present a new approach for spectral partitioning which is intended to deal with ill-posed problems in hyperspectral image classification. First, we use adaptive affinity propagation (AAP) to intelligently group the original spectral bands. Such grouping strategy not only allows us to reduce the number of spectral bands, but also to provide a different perspective on the original hyperspectral data. Then, a multiple classifier system (MCS) based on multinomial logistic regression (MLR) is applied. The system is trained using different band subsets resulting from the previously conducted intelligent grouping, and the results are combined to produce a final classification result. Our experimental results, conducted using the well-known hyperspectral scenes collected by the Airborne Visible Infra-Red Imaging Spectrometer (AVIRIS) over the Indian Pines region in NW Indiana, indicate that the proposed method can provide important advantages in terms of classification, in particular, when the number of training samples available a priori is very low.

In this letter, we propose a multinomial-logistic-regression method for pixelwise hyperspectral classification. The feature vectors are formed by the energy of the spectral vectors projected on class-indexed subspaces. In this way, we model not only the linear mixing process that is often present in the hyperspectral measurement process but also the nonlinearities that are separable in the feature space defined by the aforementioned feature vectors. Our experimental results have been conducted using both simulated and real hyperspectral data sets, which are collected using NASA's Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) and the Reflective Optics System Imaging Spectrographic (ROSIS) system. These results indicate that the proposed method provides competitive results in comparison with other state-of-the-art approaches.

This letter presents a new parallel method for hyperspectral unmixing composed by the efficient combination of two popular methods: vertex component analysis (VCA) and sparse unmixing by variable splitting and augmented Lagrangian (SUNSAL). First, VCA extracts the endmember signatures, and then, SUNSAL is used to estimate the abundance fractions. Both techniques are highly parallelizable, which significantly reduces the computing time. A design for the commodity graphics processing units of the two methods is presented and evaluated. Experimental results obtained for simulated and real hyperspectral data sets reveal speedups up to 100 times, which grants real-time response required by many remotely sensed hyperspectral applications.

Sparse unmixing has been recently introduced in hyperspectral imaging as a framework to characterize mixed pixels. It assumes that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on the ground by a field spectroradiometer). Unmixing then amounts to finding the optimal subset of signatures in a (potentially very large) spectral library that can best model each mixed pixel in the scene. In this paper, we present a refinement of the sparse unmixing methodology recently introduced which exploits the usual very low number of endmembers present in real images, out of a very large library. Specifically, we adopt the collaborative (also called “multitask” or “simultaneous”) sparse regression framework that improves the unmixing results by solving a joint sparse regression problem, where the sparsity is simultaneously imposed to all pixels in the data set. Our experimental results with both synthetic and real hyperspectral data sets show clearly the advantages obtained using the new joint sparse regression strategy, compared with the pixelwise independent approach.

Spectral unmixing aims at finding the spectrally pure constituent materials (also called endmembers) and their respective fractional abundances in each pixel of a hyperspectral image scene. In recent years, sparse unmixing has been widely used as a reliable spectral unmixing methodology. In this approach, the observed spectral vectors are expressed as linear combinations of spectral signatures assumed to be known a priori and presented in a large collection, termed spectral library or dictionary, usually acquired in laboratory. Sparse unmixing has attracted much attention as it sidesteps two common limitations of classic spectral unmixing approaches, namely, the lack of pure pixels in hyperspectral scenes and the need to estimate the number of endmembers in a given scene, which are very difficult tasks. However, the high mutual coherence of spectral libraries, jointly with their ever-growing dimensionality, strongly limits the operational applicability of sparse unmixing. In this paper, we introduce a two-step algorithm aimed at mitigating the aforementioned limitations. The algorithm exploits the usual low dimensionality of the hyperspectral data sets. The first step, which is similar to the multiple signal classification array signal processing algorithm, identifies a subset of the library elements, which contains the endmember signatures. Because this subset has cardinality much smaller than the initial number of library elements, the sparse regression we are led to is much more well conditioned than the initial one using the complete library. The second step applies collaborative sparse regression, which is a form of structured sparse regression, exploiting the fact that only a few spectral signatures in the library are active. The effectiveness of the proposed approach, termed MUSIC-CSR, is extensively validated using both simulated and real hyperspectral data sets.

Remotely sensed hyperspectral image classification is a very challenging task. This is due to many different aspects, such as the presence of mixed pixels in the data or the limited information available a priori. This has fostered the need to develop techniques able to exploit the rich spatial and spectral information present in the scenes while, at the same time, dealing with mixed pixels and limited training samples. In this paper, we present a new spectral–spatial classifier for hyperspectral data that specifically addresses the issue of mixed pixel characterization. In our presented approach, the spectral information is characterized both locally and globally, which represents an innovation with regard to previous approaches for probabilistic classification of hyperspectral data. Specifically, we use a subspace-based multinomial logistic regression method for learning the posterior probabilities and a pixel-based probabilistic support vector machine classifier as an indicator to locally determine the number of mixed components that participate in each pixel. The information provided by local and global probabilities is then fused and interpreted in order to characterize mixed pixels. Finally, spatial information is characterized by including a Markov random field (MRF) regularizer. Our experimental results, conducted using both synthetic and real hyperspectral images, indicate that the proposed classifier leads to state-of-the-art performance when compared with other approaches, particularly in scenarios in which very limited training samples are available.

In recent years, sparse representations have been widely studied in the context of remote sensing image analysis. In this paper, we propose to exploit sparse representations of morphological attribute profiles for remotely sensed image classification. Specifically, we use extended multiattribute profiles (EMAPs) to integrate the spatial and spectral information contained in the data. EMAPs provide a multilevel characterization of an image created by the sequential application of morphological attribute filters that can be used to model different kinds of structural information. Although the EMAPs' feature vectors may have high dimensionality, they lie in class-dependent low-dimensional subpaces or submanifolds. In this paper, we use the sparse representation classification framework to exploit this characteristic of the EMAPs. In short, by gathering representative samples of the low-dimensional class-dependent structures, any given sample may by sparsely represented, and thus classified, with respect to the gathered samples. Our experiments reveal that the proposed approach exploits the inherent low-dimensional structure of the EMAPs to provide state-of-the-art classification results for different multi/hyperspectral data sets.

The development of multisensor systems in recent years has led to great increase in the amount of available remote sensing data. Image fusion techniques aim at inferring high quality images of a given area from degraded versions of the same area obtained by multiple sensors. This paper focuses on pansharpening, which is the inference of a high spatial resolution multispectral image from two degraded versions with complementary spectral and spatial resolution characteristics: a) a low spatial resolution multispectral image; and b) a high spatial resolution panchromatic image. We introduce a new variational model based on spatial and spectral sparsity priors for the fusion. In the spectral domain we encourage low-rank structure, whereas in the spatial domain we promote sparsity on the local differences. Given the fact that both panchromatic and multispectral images are integrations of the underlying continuous spectra using different channel responses, we propose to exploit appropriate regularizations based on both spatial and spectral links between panchromatic and the fused multispectral images. A weighted version of the vector Total Variation (TV) norm of the data matrix is employed to align the spatial information of the fused image with that of the panchromatic image. With regard to spectral information, two different types of regularization are proposed to promote a soft constraint on the linear dependence between the panchromatic and the fused multispectral images. The first one estimates directly the linear coefficients from the observed panchromatic and low resolution multispectral images by Linear Regression (LR) while the second one employs the Principal Component Pursuit (PCP) to obtain a robust recovery of the underlying low-rank structure. We also show that the two regularizers are strongly related. The basic idea of both regularizers is that the fused image should have low-rank and preserve edge locations. We use a variation of the recently proposed Split Augmented Lagrangian Shrinkage (SALSA) algorithm to effectively solve the proposed variational formulations. Experimental results on simulated and real remote sensing images show the effectiveness of the proposed pansharpening method compared to the state-of-the-art.

This paper presents a new method to estimate the parameters of two types of blurs, linear uniform motion (approximated by a line characterized by angle and length) and out-of-focus (modeled as a uniform disk characterized by its radius), for blind restoration of natural images. The method is based on the spectrum of the blurred images and is supported on a weak assumption, which is valid for the most natural images: the power-spectrum is approximately isotropic and has a power-law decay with the spatial frequency. We introduce two modifications to the radon transform, which allow the identification of the blur spectrum pattern of the two types of blurs above mentioned. The blur parameters are identified by fitting an appropriate function that accounts separately for the natural image spectrum and the blur frequency response. The accuracy of the proposed method is validated by simulations, and the effectiveness of the proposed method is assessed by testing the algorithm on real natural blurred images and comparing it with state-of-the-art blind deconvolution methods.

In this paper, we study the separation of synchronous sources (SSS) problem, which deals with the separation of sources whose phases are synchronous. This problem cannot be addressed through independent component analysis methods because synchronous sources are statistically dependent. We present a two-step algorithm, called phase locked matrix factorization (PLMF), to perform SSS. We also show that SSS is identifiable under some assumptions and that any global minimum of PLMFs cost function is a desirable solution for SSS. We extensively study the algorithm on simulated data and conclude that it can perform SSS with various numbers of sources and sensors and with various phase lags between the sources, both in the ideal (i.e., perfectly synchronous and nonnoisy) case, and with various levels of additive noise in the observed signals and of phase jitter in the sources.

In this paper, a new parallel method for sparse spectral unmixing of remotely sensed hyperspectral data on commodity graphics processing units (GPUs) is presented. A semi-supervised approach is adopted, which relies on the increasing availability of spectral libraries of materials measured on the ground instead of resorting to endmember extraction methods. This method is based on the spectral unmixing by splitting and augmented Lagrangian (SUNSAL) that estimates the material's abundance fractions. The parallel method is performed in a pixel-by-pixel fashion and its implementation properly exploits the GPU architecture at low level, thus taking full advantage of the computational power of GPUs. Experimental results obtained for simulated and real hyperspectral datasets reveal significant speedup factors, up to 164 times, with regards to optimized serial implementation.

This paper presents a new multiple-classifier approach for accurate spectral-spatial classification of hyperspectral images, where the spectral information is exploited by combining probabilistic support vector machines (SVM) and subspace-based multinomial logistic regression (MLRsub) and the spatial information is exploited by means of a Markov random field (MRF) regularizer. The proposed approach is based on the decision fusion of global posterior probability distributions and local probabilities which result from the whole image and the class combinations map respectively. With respect to the SVM or MLRsub algorithms, the proposed method greatly improves the classification accuracy. Our experimental results with real hyperspectral images collected by the NASA Jet Propulsion Laboratory's Airborne Visible Infra-Red Imaging Spectrometer (AVIRIS) and the Reflective Optics Spectrographic Imaging System (ROSIS), indicate that the proposed multiple-classifier system leads to state-of-the-art classification performance for cases with very limited number of training samples.

In this letter, we propose a new semisupervised learning (SSL) algorithm for remotely sensed hyperspectral image classification. Our main contribution is the development of a new soft sparse multinomial logistic regression model which exploits both hard and soft labels. In our terminology, these labels respectively correspond to labeled and unlabeled training samples. The proposed algorithm represents an innovative contribution with regard to conventional SSL algorithms that only assign hard labels to unlabeled samples. The effectiveness of our proposed method is evaluated via experiments with real hyperspectral images, in which comparisons with conventional semisupervised self-learning algorithms with hard labels are carried out. In such comparisons, our method exhibits state-of-the-art performance.

In this paper, we propose a new framework for spectral-spatial classification of hyperspectral image data. The proposed approach serves as an engine in the context of which active learning algorithms can exploit both spatial and spectral information simultaneously. An important contribution of our paper is the fact that we exploit the marginal probability distribution which uses the whole information in the hyperspectral data. We learn such distributions from both the spectral and spatial information contained in the original hyperspectral data using loopy belief propagation. The adopted probabilistic model is a discriminative random field in which the association potential is a multinomial logistic regression classifier and the interaction potential is a Markov random field multilevel logistic prior. Our experimental results with hyperspectral data sets collected using the National Aeronautics and Space Administration's Airborne Visible Infrared Imaging Spectrometer and the Reflective Optics System Imaging Spectrometer system indicate that the proposed framework provides state-of-the-art performance when compared to other similar developments.

Remotely sensed hyperspectral imaging allows for the detailed analysis of the surface of the Earth using advanced imaging instruments which can produce high-dimensional images with hundreds of spectral bands. Supervised hyperspectral image classification is a difficult task due to the unbalance between the high dimensionality of the data and the limited availability of labeled training samples in real analysis scenarios. While the collection of labeled samples is generally difficult, expensive, and time-consuming, unlabeled samples can be generated in a much easier way. This observation has fostered the idea of adopting semisupervised learning techniques in hyperspectral image classification. The main assumption of such techniques is that the new (unlabeled) training samples can be obtained from a (limited) set of available labeled samples without significant effort/cost. In this paper, we develop a new approach for semisupervised learning which adapts available active learning methods (in which a trained expert actively selects unlabeled samples) to a self-learning framework in which the machine learning algorithm itself selects the most useful and informative unlabeled samples for classification purposes. In this way, the labels of the selected pixels are estimated by the classifier itself, with the advantage that no extra cost is required for labeling the selected pixels using this machine–machine framework when compared with traditional machine–human active learning. The proposed approach is illustrated with two different classifiers: multinomial logistic regression and a probabilistic pixelwise support vector machine. Our experimental results with real hyperspectral images collected by the National Aeronautics and Space Administration Jet Propulsion Laboratory's Airborne Visible–Infrared Imaging Spectrometer and the Reflective Optics Spectrographic Imaging System indicate that the use of self-learning represents an effective and promising strategy in the context of hyperspectral image classification.

This paper presents a new framework for the development of generalized composite kernel machines for hyperspectral image classification. We construct a new family of generalized composite kernels which exhibit great flexibility when combining the spectral and the spatial information contained in the hyperspectral data, without any weight parameters. The classifier adopted in this work is the multinomial logistic regression, and the spatial information is modeled from extended multiattribute profiles. In order to illustrate the good performance of the proposed framework, support vector machines are also used for evaluation purposes. Our experimental results with real hyperspectral images collected by the National Aeronautics and Space Administration Jet Propulsion Laboratory's Airborne Visible/Infrared Imaging Spectrometer and the Reflective Optics Spectrographic Imaging System indicate that the proposed framework leads to state-of-the-art classification performance in complex analysis scenarios.

In this paper, we develop a new algorithm for hyperspectral unmixing which can provide suitable endmembers (and their corresponding abundances) in a single step. Hence, the algorithm does not require a previous subspace identification step to estimate the number of endmembers as it can cope with the two most likely scenarios in practice (i.e., the number of endmembers is correctly determined or overestimated a priori). The proposed approach, termed collaborative NMF (CoNMF), uses a collaborative regularization prior which forces the abundances corresponding to the overestimated endmembers to zero, such that it is guaranteed that only the true endmembers have fractional abundance contributions and the estimation of the number of endmembers is not required in advance. The obtained experimental results demonstrate that the proposed method exhibits very good performance in case the number of endmember is not available a priori.

Vertex component analysis (VCA) has become a very popular and useful tool to linear unmix large hyperspectral datasets without the use of any a priori knowledge of the constituent spectra. Although VCA is fast method, many hyperspectral imagery applications require a response in real time or near-real time.

In this paper, we develop a new framework for semi-supervised learning which exploits active learning for unlabeled sample selection in hyperspectral data classification. Specifically, we use active learning to select the most informative unlabeled training samples with the ultimate goal of systematically achieving noticeable improvements in classification results with regard to those found by randomly selected training sets of the same size. Our experimental results, conducted with an urban hyperspectral scene collected by the Reflective Optics Spectrographic Imaging Instrument (ROSIS) of the Deutschen Zentrum for Luftund Raumfahrt (DLR, the German Aerospace Agency) over the city of Pavia, Italy, indicate that using active learning for unlabeled sample selection represents an effective and promising strategy in the context of urban hyperspectral data classification.

In our previous research, we have proposed band-similarity-based unsupervised band selection approaches, which are proven to be very efficient. In this paper, we propose to use a collaborative sparse model for further improvement. Specifically, the pre-selected bands using the fast method, called NFINDR+LP, are further refined using a collaborative sparse model. It not only requires that the linear regression coefficients are sparse, but also requires that the same set of active bands is shared by all the bands to be removed. With the collaborative sparseness constraint being relaxed, the final selected bands can be further improved, that is, the band subset with the same number of bands can provide better classification accuracy. Based on the preliminary result, the proposed sparse model is also capable of finding the minimum number of bands to be selected.

Sparse unmixing aims at estimating the constituent materials (endmembers) and their respective fractional abundances in each pixel of a hyperspectral image by assuming that the endmembers are present in a large collection of pure spectral signatures (spectral library), known a priori. In this paper, we propose a refinement of the sparse unmixing approach by taking into account the fact that all the pixels of the image share the same set of endmembers, thus lying in a lower dimensional subspace. Our idea is based on the collaborative lasso, which enforces sparsity across the pixels. The goal of this line of attack is to obtain higher accuracy of the estimated fractional abundances, at the same time with a decrease in the number of endmembers used to explain the observed data. The experimental results, obtained with both simulated and real data, confirm the potential of the proposed approach in the unmixing problem.

In this paper, we develop a new lossy compression framework for hyperspectral images, termed hyperspectral coded aperture (HYCA), which combines the ideas of spectral unmixing and compressive sensing. It takes advantage of two main properties of hyperspectral data, namely the high spatial correlation that can be observed in the data and the generally low number of endmembers needed in order to explain the data. In other words, our proposed approach intends to exploit the fact that the high dimensional hyperspectral data lives in a subspace of much lower dimension due to the mixing phenomenon. Our experimental results, conducted with synthetic hyperspectral data, indicate that the proposed approach represents a promising new strategy.

Spectral unmixing aims at estimating the fractional abundances of pure spectral signatures (also called endmembers) in each mixed pixel collected by a remote sensing hyperspectral imaging instrument. In recent work, the linear spectral unmixing problem has been approached in semisupervised fashion as a sparse regression one, under the assumption that the observed image signatures can be expressed as linear combinations of pure spectra, known a priori and available in a library. It happens, however, that sparse unmixing focuses on analyzing the hyperspectral data without incorporating spatial information. In this paper, we include the total variation (TV) regularization to the classical sparse regression formulation, thus exploiting the spatial-contextual information present in the hyperspectral images and developing a new algorithm called sparse unmixing via variable splitting augmented Lagrangian and TV. Our experimental results, conducted with both simulated and real hyperspectral data sets, indicate the potential of including spatial information (through the TV term) on sparse unmixing formulations for improved characterization of mixed pixels in hyperspectral imagery.

This paper introduces a new supervised segmentation algorithm for remotely sensed hyperspectral image data which integrates the spectral and spatial information in a Bayesian framework. A multinomial logistic regression (MLR) algorithm is first used to learn the posterior probability distributions from the spectral information, using a subspace projection method to better characterize noise and highly mixed pixels. Then, contextual information is included using a multilevel logistic Markov-Gibbs Markov random field prior. Finally, a maximum a posteriori segmentation is efficiently computed by the min-cut-based integer optimization algorithm. The proposed segmentation approach is experimentally evaluated using both simulated and real hyperspectral data sets, exhibiting state-of-the-art performance when compared with recently introduced hyperspectral image classification methods. The integration of subspace projection methods with the MLR algorithm, combined with the use of spatial-contextual information, represents an innovative contribution in the literature. This approach is shown to provide accurate characterization of hyperspectral imagery in both the spectral and the spatial domain.

This paper introduces a new unsupervised hyperspectral unmixing method conceived to linear but highly mixed hyperspectral data sets, in which the simplex of minimum volume, usually estimated by the purely geometrically based algorithms, is far way from the true simplex associated with the endmembers. The proposed method, an extension of our previous studies, resorts to the statistical framework. The abundance fraction prior is a mixture of Dirichlet densities, thus automatically enforcing the constraints on the abundance fractions imposed by the acquisition process, namely, nonnegativity and sum-to-one. A cyclic minimization algorithm is developed where the following are observed: 1) The number of Dirichlet modes is inferred based on the minimum description length principle; 2) a generalized expectation maximization algorithm is derived to infer the model parameters; and 3) a sequence of augmented Lagrangian-based optimizations is used to compute the signatures of the endmembers. Experiments on simulated and real data are presented to show the effectiveness of the proposed algorithm in unmixing problems beyond the reach of the geometrically based state-of-the-art competitors.

Augmented Lagrangian variational formulations and alternating optimization have been adopted to solve distributed parameter estimation problems. The alternating direction method of multipliers (ADMM) is one of such formulations/optimization methods. Very recently, the number of applications of the ADMM, or variants of it, to solve inverse problems in image and signal processing has increased at an exponential rate. The reason for this interest is that ADMM decomposes a difficult optimization problem into a sequence of much simpler problems. In this paper, we use the ADMM to reconstruct piecewise-smooth distributed parameters of elliptical partial differential equations from noisy and linear (blurred) observations of the underlying field. The distributed parameters are estimated by solving an inverse problem with total variation (TV) regularization. The proposed instance of the ADMM solves, in each iteration, an l(2) and a decoupled l(2) - l(1) optimization problems. An operator splitting is used to simplify the treatment of the TV regularizer, avoiding its smooth approximation and yielding a simple yet effective ADMM reconstruction method compared with previously proposed approaches. The competitiveness of the proposed method, with respect to the state-of-the-art, is illustrated in simulated 1-D and 2-D elliptical equation problems, which are representative of many real applications.

Hyperspectral instruments acquire electromagnetic energy scattered within their ground instantaneous field view in hundreds of spectral channels with high spectral resolution. Very often, however, owing to low spatial resolution of the scanner or to the presence of intimate mixtures (mixing of the materials at a very small scale) in the scene, the spectral vectors (collection of signals acquired at different spectral bands from a given pixel) acquired by the hyperspectral scanners are actually mixtures of the spectral signatures of the materials present in the scene. Given a set of mixed spectral vectors, spectral mixture analysis (or spectral unmixing) aims at estimating the number of reference materials, also called endmembers, their spectral signatures, and their fractional abundances. Spectral unmixing is, thus, a source separation problem. This paper presents an overview of the principal research directions in hyperspectral unmixing. The paper is organized into six main topics: i) mixing models, ii) signal subspace identification, iii) geometrical-based spectral unmixing, iv) statistical-based spectral unmixing, v) sparse regression based unmixing, and vi) spatial-contextual information. For each topic, we summarize what is the mathematical problem involved and give relevant pointers to state-of-the-art algorithms to address these problems.

Sparse unmixing has been recently introduced as a mechanism to characterize mixed pixels in remotely sensed hyper-spectral images. It assumes that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on the ground by a field spectroradiometer). Unmixing then amounts to finding the optimal subset of signatures in a (potentially very large) spectral library that can best model each mixed pixel in the scene. In available spectral libraries, it is observed that the spectral signatures appear organized in groups (e.g. different alterations of a single mineral in the U.S. Geological Survey spectral library). In this paper, we explore the potential of the sparse group lasso technique in solving hyperspectral unmixing problems. Our introspection in this work is that, when the spectral signatures appear in groups, this technique has the potential to yield better results than the standard sparse regression approach. Experimental results with both synthetic and real hyperspectral data are given to investigate this issue.

In this work, we present a new subspace discriminant analysis classification algorithm for remotely sensed hyperspectral image data. Our motivation for including subspace projection as a distinctive feature of our work is to better model noise and mixed pixels present in hyperspectral images. Two different dimensionality reduction techniques are considered: principal component analysis (PCA) and the hyperspectral signal identification by minimum error (HySime) algorithm. Experimental results indicate that the proposed method can provide competitive classification results (in the presence of very limited training data sets) with regards to those achieved by other state-of-the-art methods, such as linear discriminant analysis (LDA), subspace LDA, support vector machines (SVMs), and subspace SVMs using PCA and HySime for dimensionality reduction purposes.

This paper introduces a new supervised technique to segment hyperspectral images: the Bayesian segmentation based on discriminative classification and on multilevel logistic (MLL) spatial prior. The approach is Bayesian and exploits both spectral and spatial information. Given a spectral vector, the posterior class probability distribution is modeled using multinomial logistic regression (MLR) which, being a discriminative model, allows to learn directly the boundaries between the decision regions and, thus, to successfully deal with high-dimensionality data. To control the machine complexity and, thus, its generalization capacity, the prior on the multinomial logistic vector is assumed to follow a componentwise independent Laplacian density. The vector of weights is computed via the fast sparse multinomial logistic regression (FSMLR), a variation of the sparse multinomial logistic regression (SMLR), conceived to deal with large data sets beyond the reach of the SMLR. To avoid the high computational complexity involved in estimating the Laplacian regularization parameter, we have also considered the Jeffreys prior, as it does not depend on any hyperparameter. The prior probability distribution on the class-label image is an MLL Markov–Gibbs distribution, which promotes segmentation results with equal neighboring class labels. The$\alpha$-expansion optimization algorithm, a powerful graph-cut-based integer optimization tool, is used to compute the maximum a posteriori segmentation. The effectiveness of the proposed methodology is illustrated by comparing its performance with the state-of-the-art methods on synthetic and real hyperspectral image data sets. The reported results give clear evidence of the relevance of using both spatial and spectral information in hyperspectral image segmentation.

Linear spectral unmixing is a popular tool in remotely sensed hyperspectral data interpretation. It aims at estimating the fractional abundances of pure spectral signatures (also called as endmembers) in each mixed pixel collected by an imaging spectrometer. In many situations, the identification of the end-member signatures in the original data set may be challenging due to insufficient spatial resolution, mixtures happening at different scales, and unavailability of completely pure spectral signatures in the scene. However, the unmixing problem can also be approached in semisupervised fashion, i.e., by assuming that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance (e.g., spectra collected on the ground by a field spectroradiometer). Unmixing then amounts to finding the optimal subset of signatures in a (potentially very large) spectral library that can best model each mixed pixel in the scene. In practice, this is a combinatorial problem which calls for efficient linear sparse regression (SR) techniques based on sparsity-inducing regularizers, since the number of endmembers participating in a mixed pixel is usually very small compared with the (ever-growing) dimensionality (and availability) of spectral libraries. Linear SR is an area of very active research, with strong links to compressed sensing, basis pursuit (BP), BP denoising, and matching pursuit. In this paper, we study the linear spectral unmixing problem under the light of recent theoretical results published in those referred to areas. Furthermore, we provide a comparison of several available and new linear SR algorithms, with the ultimate goal of analyzing their potential in solving the spectral unmixing problem by resorting to available spectral libraries. Our experimental results, conducted using both simulated and real hyperspectral data sets collected by the NASA Jet Propulsion Laboratory's Airborne Visible Infrared Imaging Spectrometer and spectral libraries publicly available from the U.S. Geological Survey, indicate the potential of SR techniques in the task of accurately characterizing the mixed pixels using the library spectra. This opens new perspectives for spectral unmixing, since the abundance estimation process no longer depends on the availability of pure spectral signatures in the input data nor on the capacity of a certain endmember extraction algorithm to identify such pure signatures.

This paper introduces a new supervised Bayesian approach to hyperspectral image segmentation with active learning, which consists of two main steps. First, we use a multinomial logistic regression (MLR) model to learn the class posterior probability distributions. This is done by using a recently introduced logistic regression via splitting and augmented Lagrangian algorithm. Second, we use the information acquired in the previous step to segment the hyperspectral image using a multilevel logistic prior that encodes the spatial information. In order to reduce the cost of acquiring large training sets, active learning is performed based on the MLR posterior probabilities. Another contribution of this paper is the introduction of a new active sampling approach, called modified breaking ties, which is able to provide an unbiased sampling. Furthermore, we have implemented our proposed method in an efficient way. For instance, in order to obtain the time-consuming maximum a posteriori segmentation, we use the α-expansion min-cut-based integer optimization algorithm. The state-of-the-art performance of the proposed approach is illustrated using both simulated and real hyperspectral data sets in a number of experimental comparisons with recently introduced hyperspectral image analysis methods.

The 19 papers in this special issue focus on the state-of-the-art and most recent developments in the area of spectral unmixing of remotely sensed data.

We propose a new fast algorithm for solving one of the standard approaches to ill-posed linear inverse problems (IPLIP), where a (possibly nonsmooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and constraint are usually convex, several particular features of these problems (huge dimensionality, nonsmoothness) preclude the use of off-the-shelf optimization tools and have stimulated a considerable amount of research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the family of augmented Lagrangian methods, can be used to deal with a variety of imaging IPLIP, including deconvolution and reconstruction from compressive observations (such as MRI), using either total-variation or wavelet-based (or, more generally, frame-based) regularization. The proposed algorithm is an instance of the so-called alternating direction method of multipliers, for which convergence sufficient conditions are known; we show that these conditions are satisfied by the proposed algorithm. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is a strong contender for the state-of-the-art.

It has been proven that there are synchrony (or phase-locking) phenomena present in multiple oscillating systems such as electrical circuits, lasers, chemical reactions, and human neurons. If the measurements of these systems cannot detect the individual oscillators but rather a superposition of them, as in brain electrophysiological signals (electro- and magneoencephalogram), spurious phase locking will be detected. Current source-extraction techniques attempt to undo this superposition by assuming properties on the data, which are not valid when underlying sources are phase-locked. Statistical independence of the sources is one such invalid assumption, as phase-locked sources are dependent. In this paper, we introduce methods for source separation and clustering which make adequate assumptions for data where synchrony is present, and show with simulated data that they perform well even in cases where independent component analysis and other well-known source-separation methods fail. The results in this paper provide a proof of concept that synchrony-based techniques are useful for low-noise applications.

Ill-posed linear inverse problems (ILIP), such as restoration and reconstruction, are a core topic of signal/image processing. A standard formulation for dealing with ILIP consists in a constrained optimization problem, where a regularization function is minimized under the constraint that the solution explains the observations sufficiently well. The regularizer and constraint are usually convex; however, several particular features of these problems (huge dimensionality, non-smoothness) preclude the use of off-the-shelf optimization tools and have stimulated much research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the category of augmented Lagrangian methods, can be used to deal with a variety of imaging ILIP, including deconvolution and reconstruction from compressive observations (such as MRI). Experiments testify for the effectiveness of the proposed method.

In some imaging inverse problems, it may be desired that the solution simultaneously exhibits a set of properties not enforceable by a single regularizer. To attain this goal, one may use a linear combinations of regularizers, thus encouraging the solution to simultaneously exhibit the characteristics enforced by each of them. This paper addresses the optimization problem associated with this type of compound regularization, using an alternating direction optimization algorithm. We illustrate the approach in two image deblurring problems - one in which the images are simultaneously sparse and piece-wise smooth, using a linear combination of the ℓ1and total variation regularizers, and the other for a natural image with a combination of frame-based synthesis and analysis ℓ1norm regularizers.

Restoration of Poissonian images is a class of inverse problem arising in fields such medical and astronomical imaging. Regularization criteria that combine the Poisson log-likelihood with a non-smooth convex regularizer lead to optimization problems with several difficulties: the log-likelihood does not have a Lipschitzian gradient; the regularizer is non-smooth; there is a non-negativity constraint. Using convex analysis tools, we give sufficient conditions for existence and uniqueness of solutions of these optimization problems for (frame-based) analysis and synthesis formulations. Then, we attack these problems with an adapted version of the alternating direction method of multipliers and show that sufficient conditions for convergence are met. The algorithm is shown to be competitive, often outperform, state-of-the-art methods.

This paper explores the applicability of new sparse algorithms to perform spectral unmixing of hyperspectral images using available spectral libraries instead of resorting to well-known end member extraction techniques widely available in the literature. Our main assumption is that it is unlikely to find pure pixels in real hyperspectral images due to available spatial resolution and mixing phenomena happening at different scales. The algorithms analyzed in our study rely on different principles, and their performance is quantitatively assessed using both simulated and real hyperspectral data sets. The experimental validation of sparse techniques conducted in this work indicates promising results of this new approach to attack the spectral unmixing problem in remotely sensed hyperspectral images.

Land surface temperature (LST) is estimated from thermal infrared data provided by the Spinning Enhanced Visible and Infrared Imager (SEVIRI) onboard Meteosat Second Generation (MSG), using a generalized split-window (GSW) algorithm. The uncertainty of the LST retrievals is highly dependent on the input accuracy and retrieval conditions, particularly the sensor view angle and the atmospheric water vapor content. This paper presents a quantification of the uncertainty of LST estimations, taking into account error statistics of the GSW under a globally representative collection of atmospheric profiles, and a careful characterization of the uncertainty of input data, particularly the surface emissivity and forecasts of the total water vapor content. Such analysis is the basis for LST uncertainty estimation, also distributed to users, in the form of error bars, along with the LST retrievals. Moreover, the spatial coverage of SEVIRI LST is essentially determined by the LST expected uncertainty, instead of being restricted to view zenith angles below a given threshold (e.g., 60°). Within the MSG disk, the atmosphere is often dry for clear-sky conditions where angles are large (e.g., Northern and Eastern Europe and Saudi Arabia). By considering several factors that contribute to LST inaccuracies, it is possible to increase the spatial coverage to regions such as those mentioned earlier. Retrieved values are also compared within situobservations collected in Namibia, covering a seasonal cycle. The two data sets are in good agreement with root-mean-square differences ranging between 1°C and 2°C, which is well below the average error estimated for the satellite retrievals.

This paper presents a new semisupervised segmentation algorithm, suited to high-dimensional data, of which remotely sensed hyperspectral image data sets are an example. The algorithm implements two main steps: 1) semisupervised learning of the posterior class distributions followed by 2) segmentation, which infers an image of class labels from a posterior distribution built on the learned class distributions and on a Markov random field. The posterior class distributions are modeled using multinomial logistic regression, where the regressors are learned using both labeled and, through a graph-based technique, unlabeled samples. Such unlabeled samples are actively selected based on the entropy of the corresponding class label. The prior on the image of labels is a multilevel logistic model, which enforces segmentation results in which neighboring labels belong to the same class. The maximum a posteriori segmentation is computed by the α-expansion min-cut-based integer optimization algorithm. Our experimental results, conducted using synthetic and real hyperspectral image data sets collected by the Airborne Visible/Infrared Imaging Spectrometer system of the National Aeronautics and Space Administration Jet Propulsion Laboratory over the regions of Indian Pines, IN, and Salinas Valley, CA, reveal that the proposed approach can provide classification accuracies that are similar or higher than those achieved by other supervised methods for the considered scenes. Our results also indicate that the use of a spatial prior can greatly improve the final results with respect to a case in which only the learned class densities are considered, confirming the importance of jointly considering spatial and spectral information in hyperspectral image segmentation.

We propose a new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an l2 data-fidelity term and a nonsmooth regularizer. This formulation allows both wavelet-based (with orthogonal or frame-based representations) regularization or total-variation regularization. Our approach is based on a variable splitting to obtain an equivalent constrained optimization formulation, which is then addressed with an augmented Lagrangian method. The proposed algorithm is an instance of the so-called alternating direction method of multipliers, for which convergence has been proved. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is faster than the current state of the art methods.

Multiplicative noise (also known as speckle noise) models are central to the study of coherent imaging systems, such as synthetic aperture radar and sonar, and ultrasound and laser imaging. These models introduce two additional layers of difficulties with respect to the standard Gaussian additive noise scenario: (1) the noise is multiplied by (rather than added to) the original image; (2) the noise is not Gaussian, with Rayleigh and Gamma being commonly used densities. These two features of multiplicative noise models preclude the direct application of most state-of-the-art algorithms, which are designed for solving unconstrained optimization problems where the objective has two terms: a quadratic data term (log-likelihood), reflecting the additive and Gaussian nature of the noise, plus a convex (possibly nonsmooth) regularizer (e.g., a total variation or wavelet-based regularizer/prior). In this paper, we address these difficulties by: (1) converting the multiplicative model into an additive one by taking logarithms, as proposed by some other authors; (2) using variable splitting to obtain an equivalent constrained problem; and (3) dealing with this optimization problem using the augmented Lagrangian framework. A set of experiments shows that the proposed method, which we name MIDAL (multiplicative image denoising by augmented Lagrangian), yields state-of-the-art results both in terms of speed and denoising performance.

Much research has been devoted to the problem of restoring Poissonian images, namely for medical and astronomical applications. However, the restoration of these images using state-of-the-art regularizers (such as those based upon multiscale representations or total variation) is still an active research area, since the associated optimization problems are quite challenging. In this paper, we propose an approach to deconvolving Poissonian images, which is based upon an alternating direction optimization method. The standard regularization [or maximum a posteriori (MAP)] restoration criterion, which combines the Poisson log-likelihood with a (nonsmooth) convex regularizer (log-prior), leads to hard optimization problems: the log-likelihood is nonquadratic and nonseparable, the regularizer is nonsmooth, and there is a nonnegativity constraint. Using standard convex analysis tools, we present sufficient conditions for existence and uniqueness of solutions of these optimization problems, for several types of regularizers: total-variation, frame-based analysis, and frame-based synthesis. We attack these problems with an instance of the alternating direction method of multipliers (ADMM), which belongs to the family of augmented Lagrangian algorithms. We study sufficient conditions for convergence and show that these are satisfied, either under total-variation or frame-based (analysis and synthesis) regularization. The resulting algorithms are shown to outperform alternative state-of-the-art methods, both in terms of speed and restoration accuracy.

Multiplicative noise models occur in the study of several coherent imaging systems, such as synthetic aperture radar and sonar, and ultrasound and laser imaging. This type of noise is also commonly referred to as speckle. Multiplicative noise introduces two additional layers of difficulties with respect to the popular Gaussian additive noise model: (1) the noise is multiplied by (rather than added to) the original image, and (2) the noise is not Gaussian, with Rayleigh and Gamma being commonly used densities. These two features of the multiplicative noise model preclude the direct application of state-of-the-art restoration methods, such as those based on the combination of total variation or wavelet-based regularization with a quadratic observation term. In this paper, we tackle these difficulties by: (1) using the common trick of converting the multiplicative model into an additive one by taking logarithms, and (2) adopting the recently proposed split Bregman approach to estimate the underlying image under total variation regularization. This approach is based on formulating a constrained problem equivalent to the original unconstrained one, which is then solved using Bregman iterations (equivalently, an augmented Lagrangian method). A set of experiments show that the proposed method yields state-of-the-art results.

Finding an accurate sparse approximation of a spectral vector described by a linear model, when there is available a library of possible constituent signals (called endmembers or atoms), is a hard combinatorial problem which, as in other areas, has been increasingly addressed. This paper studies the efficiency of the sparse regression techniques in the spectral unmixing problem by conducting a comparison between four different approaches: Moore-Penrose Pseudoinverse, Orthogonal Matching Pursuit (OMP), Iterative Spectral Mixture Analysis (ISMA) and l2- l1sparse regression techniques, which are of widespread use in compressed sensing. We conclude that the l2-l1sparse regression techniques, implemented here by Iterative Shrinkage/Thresholding (TwIST) algorithm, yield the state-of-the-art in the hyperspectral unmixing area.

This paper introduces a new semi-supervised classification and segmentation approach tailored to hyperspectral images. The posterior distributions of the classes are modeled by the multinomial logistic regression. The contextual information inherent to the spatial configuration of the image pixels is modeled by a Multi-Level Logistic (MLL) Markov-Gibbs random field. The multinomial logistic regressors, assumed to be random vectors with independent Lapla-cian components, are learned using the recently introduced LOR-SAL algorithm. The maximum a posteriori (MAP) segmentation is computed via the α-Expansion algorithm, a powerful graph cut based approach to integer optimization. The effectiveness of the proposed methodology is illustrated by classifying simulated and real data sets. Comparisons with state-of-art methods are also included.

In several imaging inverse problems, it may be of interest to encourage the solution to have characteristics which are most naturally expressed by the combination of more than one regularizer. The resulting optimization problems can not be dealt with by the current state-of-the-art algorithms, which are designed for single regularizers (such as total variation or sparseness-inducing penalties, but not both simultaneously). In this paper, we introduce an iterative algorithm to solve the optimization problem resulting from image (or signal) inverse problems with two (or more) regularizers. We illustrate the new algorithm in a problem of restoration of "group sparse" images, i.e., images displaying a special type of sparseness in which the active pixels tend to cluster together. Experimental results show the effectiveness of the proposed algorithm in solving the corresponding optimization problem.

Medical images are often noisy owing to the physical mechanisms of the acquisition process. The great majority of the denoising algorithms assume additive white Gaussian noise. However, some of the most popular medical image modalities are degraded by some type of non-Gaussian noise. Among these types, we refer the Poisson noise, which is particularly suitable for modeling the counting processes associated to many imaging modalities such as PET, SPECT, and fluorescent confocal microscopy imaging. The aim of this work is to compare the effectiveness of several denoising algorithms in the presence of Poisson noise. We consider algorithms specifically designed for Poisson noise (wavelets, Platelets, and minimum descritpion length) and algorithms designed for Gaussian noise (edge preserving bilateral filtering, total variation, and non-local means). These algorithms are applied to piecewise smooth simulated and real data. Somehow unexpectedly, we conclude that total variation, designed for Gaussian noise, outperforms more elaborated state-of-the-art methods specifically designed for Poisson noise.

This paper presents a new method of minimum volume class for hyperspectral unmixing, termed minimum volume simplex analysis (MVSA). The underlying mixing model is linear; i.e., the mixed hyperspectral vectors are modeled by a linear mixture of the endmember signatures weighted by the correspondent abundance fractions. MVSA approaches hyperspectral unmixing by fitting a minimum volume simplex to the hyperspectral data, constraining the abundance fractions to belong to the probability simplex. The resulting optimization problem is solved by implementing a sequence of quadratically constrained subproblems. In a final step, the hard constraint on the abundance fractions is replaced with a hinge type loss function to account for outliers and noise. We illustrate the state-of-the-art performance of the MVSA algorithm in unmixing simulated data sets. We are mainly concerned with the realistic scenario in which the pure pixel assumption (i.e., there exists at least one pure pixel per endmember) is not fulfilled. In these conditions, the MVSA yields much better performance than the pure pixel based algorithms.

Signal subspace identification is a crucial first step in many hyperspectral processing algorithms such as target detection, change detection, classification, and unmixing. The identification of this subspace enables a correct dimensionality reduction, yielding gains in algorithm performance and complexity and in data storage. This paper introduces a new minimum mean square error-based approach to infer the signal subspace in hyperspectral imagery. The method, which is termed hyperspectral signal identification by minimum error, is eigen decomposition based, unsupervised, and fully automatic (i.e., it does not depend on any tuning parameters). It first estimates the signal and noise correlation matrices and then selects the subset of eigenvalues that best represents the signal subspace in the least squared error sense. State-of-the-art performance of the proposed method is illustrated by using simulated and real hyperspectral images.

Iterative shrinkage/thresholding (IST) algorithms have been recently proposed to handle high-dimensional convex optimization problems arising in image inverse problems (namely deconvolution) under non-quadratic regularization (e.g., total variation or sparsity inducing regularizers on wavelet representations). The convergence speed of IST algorithms depends heavily on the nature of the direct operator, being very slow when this operator is severely ill-conditioned. In this paper, we introduce a two-step version of IST (termed 2IST, pronounced "twist") showing much faster convergence for strongly ill-conditioned operators. We give theoretical results concerning the convergence behavior of 2IST and show its effectiveness for wavelet-based and total variation image deconvolution.

A Bayesian segmentation approach for hyperspectral images is introduced in this paper. The method improves the classification performance of discriminative classifiers by adding contextual information in the form of spatial dependencies. The technique herein presented builds the class densities based on fast sparse multinomial logistic regression and enforces spacial continuity by adopting a multi-level logistic Markov-Gibs prior. State-of-art performance of the proposed approach is illustrated in a set of experimental comparisons with recently introduced hyperspectral classification/segmentation methods.

Given an hyperspectral image, the determination of the number of endmembers and the subspace where they live without any prior knowledge is crucial to the success of hyperspectral image analysis. This paper introduces a new minimum mean squared error based approach to infer the signal subspace in hyperspectral imagery. The method, termed hyperspectral signal identification by minimum error (HySime), is eigendecomposition based and it does not depend on any tuning parameters. It first estimates the signal and noise correlation matrices and then selects the subset of eigenvalues that best represents the signal subspace in the least squared error sense. The effectiveness of the proposed method is illustrated using simulated data based on U.S.G.S. laboratory spectra and real hyperspectral data collected by the AVIRIS sensor over Cuprite, Nevada.

This paper introduces a new method to blindly unmix hyperspectral data, termed dependent component analysis (DECA). This method decomposes a hyperspectral images into a collection of reflectance (or radiance) spectra of the materials present in the scene (end member signatures) and the corresponding abundance fractions at each pixel. DECA assumes that each pixel is a linear mixture of the end-members signatures weighted by the correspondent abundance fractions. These abundances are modeled as mixtures of Dirichlet densities, thus enforcing the constraints on abundance fractions imposed by the acquisition process, namely non-negativity and constant sum. The mixing matrix is inferred by a generalized expectation-maximization (GEM) type algorithm. This method overcomes the limitations of unmixing methods based on independent component analysis (ICA) and on geometrical based approaches. The effectiveness of the proposed method is illustrated using simulated data based on U.S.G.S. laboratory spectra and real hyperspectral data collected by the AVIRIS sensor over Cuprite, Nevada.

Segmentation of dark patches in SAR images is an important step in any oil spill detection system. Segmentation methods used so far include 'adaptive image thresholding', 'hysteresis thresholding', 'edge detection' (see [1] and references therein) and entropy methods like the 'maximum descriptive length' technique [2]. This paper extends and generalizes a previously proposed Bayesian semi-supervised segmentation algorithm [3] oriented to oil spill detection using SAR images. In the base algorithm on which we build on, the data term is modeled by a finite mixture of Gamma distributions, with a given predefined number of components, for modeling each one of two classes (oil and water). To estimate the parameters of the class conditional densities, an expectation maximization (EM) algorithm was developed. The prior is an M- level logistic (MLL) Markov Random Field enforcing local continuity in a statistical sense. The methodology proposed in [3] assumes two classes and known smoothness parameter. The present work removes these restrictions. The smoothness parameter controlling the degree of homogeneity imposed on the scene is automatically estimated and the number of used classes is optional. To extend the algorithm to an optional number of classes, the so-called alpha-expansion algorithm [4] has been implemented. This algorithm is a graph-cut based technique that finds efficiently (polynomial complexity) the local minimum of the energy, (i.e, a labeling) within a known factor of the global minimum. In order to estimate the smoothness parameter of the MLL prior, two different techniques have been tested, namely the least squares (LS) fit and the coding method (CD) [5]. Semi-automatic estimation of the class parameters is also implemented. This represents an improvement over the base algorithm [3], where parameter estimation is performed on a supervised way by requesting user defined regions of interest representing the water and the oil. The effectiveness of the proposed approach is illustrated with simulated SAR images and real ERS and ENVISAT images.

Abstract—This paper addresses the main goals and objec-tives of the Hyperspectral Imaging Network (HYPER-I-NET), a recently started Marie Curie Research Training Network. The project is designed to build an interdisciplinary research community focusing on hyperspectral imaging activities. The core strategy of the network is to create a powerful interdisciplinary synergy between different domains of expertise closely related to hyperspectral imaging activities in Europe, ranging from sensor design and flight operation to data collection, processing, interpretation, and dissemination. Our main goals in this paper are to present the project to the Geoscience and Remote Sensing community and to provide an overview of the planned activities in each sub-activity covered by the network.

Iterative shrinkage/thresholding (IST) algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems. This class of problems results from combining a linear observation model with a nonquadratic regularizer (e.g., total variation or wavelet-based regularization). It happens that the convergence rate of these IST algorithms depends heavily on the linear observation operator, becoming very slow when this operator is ill-conditioned or ill-posed. In this paper, we introduce two-step IST (TwIST) algorithms, exhibiting much faster convergence rate than IST for ill-conditioned problems. For a vast class of nonquadratic convex regularizers (l(p) norms, some Besov norms, and total variation), we show that TwIST converges to a minimizer of the objective function, for a given range of values of its parameters. For noninvertible observation operators, we introduce a monotonic version of TwIST (MTwIST); although the convergence proof does not apply to this scenario, we give experimental evidence that MTwIST exhibits similar speed gains over IST. The effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples.

Phase unwrapping is the inference of absolute phase from modulo-2pi phase. This paper introduces a new energy minimization framework for phase unwrapping. The considered objective functions are first-order Markov random fields. We provide an exact energy minimization algorithm, whenever the corresponding clique potentials are convex, namely for the phase unwrapping classical Lp norm, with p > or = 1. Its complexity is KT (n, 3n), where K is the length of the absolute phase domain measured in 2pi units and T (n, m) is the complexity of a max-flow computation in a graph with n nodes and m edges. For nonconvex clique potentials, often used owing to their discontinuity preserving ability, we face an NP-hard problem for which we devise an approximate solution. Both algorithms solve integer optimization problems by computing a sequence of binary optimizations, each one solved by graph cut techniques. Accordingly, we name the two algorithms PUMA, for phase unwrappping max-flow/min-cut. A set of experimental results illustrates the effectiveness of the proposed approach and its competitiveness in comparison with state-of-the-art phase unwrapping algorithms.

Standard formulations of image/signal deconvolution under wavelet-based priors/regularizers lead to very high-dimensional optimization problems involving the following difficulties: the non-Gaussian (heavy-tailed) wavelet priors lead to objective functions which are nonquadratic, usually nondifferentiable, and sometimes even nonconvex; the presence of the convolution operator destroys the separability which underlies the simplicity of wavelet-based denoising. This paper presents a unified view of several recently proposed algorithms for handling this class of optimization problems, placing them in a common majorization-minimization (MM) framework. One of the classes of algorithms considered (when using quadratic bounds on nondifferentiable log-priors) shares the infamous "singularity issue" (SI) of "iteratively reweighted least squares" (IRLS) algorithms: the possibility of having to handle infinite weights, which may cause both numerical and convergence issues. In this paper, we prove several new results which strongly support the claim that the SI does not compromise the usefulness of this class of algorithms. Exploiting the unified MM perspective, we introduce a new algorithm, resulting from using l1 bounds for nonconvex regularizers; the experiments confirm the superior performance of this method, when compared to the one based on quadratic majorization. Finally, an experimental comparison of the several algorithms, reveals their relative merits for different standard types of scenarios.

The total variation regularizer is well suited to piecewise smooth images. If we add the fact that these regularizers are convex, we have, perhaps, the reason for the resurgence of interest on TV-based approaches to inverse problems. This paper proposes a new TV-based algorithm for image deconvolution, under the assumptions of linear observations and additive white Gaussian noise. To compute the TV estimate, we propose a majorization-minimization approach, which consists in replacing a difficult optimization problem by a sequence of simpler ones, by relying on convexity arguments. The resulting algorithm has O(N) computational complexity, for finite support convolutional kernels. In a comparison with state-of-the-art methods, the proposed algorithm either outperforms or equals them, with similar computational complexity

Image denoising is a classical problem which has been addressed using a variety of conceptual frameworks and computational tools. Most approaches use some form of penalty/prior as a regularizer, expressing a preference for images with some form of (generalized) "smoothness". Total variation (TV) and wavelet-based methods have received a great deal of attention in the last decade and are among the state of the art in this problem. However, as far as we know, no experimental studies have been carried out, comparing the relative performance of the two classes of methods. In this paper, we present the results of such a comparison. Prior to that, we introduce a new majorization-minimization algorithm to implement the TV denoising criterion. We conclude that TV is outperformed by recent state of the art wavelet-based denoising methods, but performs competitively with older wavelet-based methods.

Image deconvolution is formulated in the wavelet domain under the Bayesian framework. The well-known sparsity of the wavelet coefficients of real-world images is modeled by heavy-tailed priors belonging to the Gaussian scale mixture (GSM) class; i.e., priors given by a linear (finite of infinite) combination of Gaussian densities. This class includes, among others, the generalized Gaussian, the Jeffreys, and the Gaussian mixture priors. Necessary and sufficient conditions are stated under which the prior induced by a thresholding/shrinking denoising rule is a GSM. This result is then used to show that the prior induced by the "nonnegative garrote" thresholding/shrinking rule, herein termed the garrote prior, is a GSM. To compute the maximum a posteriori estimate, we propose a new generalized expectation maximization (GEM) algorithm, where the missing variables are the scale factors of the GSM densities. The maximization step of the underlying expectation maximization algorithm is replaced with a linear stationary second-order iterative method. The result is a GEM algorithm of O(N log N) computational complexity. In a series of benchmark tests, the proposed approach outperforms or performs similarly to state-of-the art methods, demanding comparable (in some cases, much less) computational complexity.

The paper presents a new max-flow/min-cut approach for recovering the absolute phase from modulo-2/spl pi/ phase, the so-called phase unwrapping (PU) problem. The adopted criterion is the minimization of the L/sup p/ norm of phase differences, leading to computationally demanding integer optimization problems. The unwrapped phase is computed iteratively through a sequence of binary optimizations, each one mapped onto a max-flow/min-cut problem on a certain graph. Accordingly, we name this new algorithm PUMF (for phase unwrapping max-flow). A set of experimental results illustrates the effectiveness of PUMF approach, namely for 0 < p < 1, where most competitors fail.

The paper proposes a Bayesian 3D ultrasound reconstruction/estimation from non-uniform ultrasound observations, non-Gaussian data fidelity term, and total variation (TV) based prior. To compute the maximum a posteriori (MAP) solution, we introduce a generalized expectation maximization (GEM) algorithm, which converges to the exact MAP solution in the case of convex data fidelity term. A set of experiments illustrates the effectiveness of the method.

Independent component analysis (ICA) has recently been proposed as a tool to unmix hyperspectral data. ICA is founded on two assumptions: 1) the observed spectrum vector is a linear mixture of the constituent spectra (endmember spectra) weighted by the correspondent abundance fractions (sources); 2)sources are statistically independent. Independent factor analysis (IFA) extends ICA to linear mixtures of independent sources immersed in noise. Concerning hyperspectral data, the first assumption is valid whenever the multiple scattering among the distinct constituent substances (endmembers) is negligible, and the surface is partitioned according to the fractional abundances. The second assumption, however, is violated, since the sum of abundance fractions associated to each pixel is constant due to physical constraints in the data acquisition process. Thus, sources cannot be statistically independent, this compromising the performance of ICA/IFA algorithms in hyperspectral unmixing. This paper studies the impact of hyperspectral source statistical dependence on ICA and IFA performances. We conclude that the accuracy of these methods tends to improve with the increase of the signature variability, of the number of endmembers, and of the signal-to-noise ratio. In any case, there are always endmembers incorrectly unmixed. We arrive to this conclusion by minimizing the mutual information of simulated and real hyperspectral mixtures. The computation of mutual information is based on fitting mixtures of Gaussians to the observed data. A method to sort ICA and IFA estimates in terms of the likelihood of being correctly unmixed is proposed.

Given a set of mixed spectral (multispectral or hyperspectral) vectors, linear spectral mixture analysis, or linear unmixing, aims at estimating the number of reference substances, also called endmembers, their spectral signatures, and their abundance fractions. This paper presents a new method for unsupervised endmember extraction from hyperspectral data, termed vertex component analysis (VCA). The algorithm exploits two facts: (1) the endmembers are the vertices of a simplex and (2) the affine transformation of a simplex is also a simplex. In a series of experiments using simulated and real data, the VCA algorithm competes with state-of-the-art methods, with a computational complexity between one and two orders of magnitude lower than the best available method.

Reconstruction of the backscatter and extinction coefficients is a crucial step in many quantitative remote sensing applications, such as radar, light detection and ranging (lidar), and sonar. We present a novel stochastic filtering approach for the estimation of the backscatter and extinction coefficients from time-range elastic-backscatter lidar data. The Bayesian perspective is adopted; we take as prior a causal first-order autoregressive Gauss-Markov random field tailored to enforce smoothness on time and range dimensions. By using a reduced-order state-space representation of the prior, we derive a suboptimal stochastic filter that recursively computes the backscatter and extinction coefficients at each range-time inversion cell. The estimator is a kind of adaptive extended Kalman filter, being efficient from the computational point of view. A set of experiments illustrates the effectiveness of the proposed approach, namely its advantage over the classical Klett deterministic approach.

The paper proposes a new wavelet-based Bayesian approach to image deconvolution, under the space-invariant blur and additive white Gaussian noise assumptions. Image deconvolution exploits the well known sparsity of the wavelet coefficients, described by heavy-tailed priors. The present approach admits any prior given by a linear (finite of infinite) combination of Gaussian densities. To compute the maximum a posteriori (MAP) estimate, we propose a generalized expectation maximization (GEM) algorithm where the missing variables are the Gaussian modes. The maximization step of the EM algorithm is approximated by a stationary second order iterative method. The result is a GEM algorithm of O(N log N) computational complexity. In comparison with state-of-the-art methods, the proposed algorithm either outperforms or equals them, with low computational complexity.

An efficient algorithm for the computation of tr{TR/sup -1/}, where T and R are Toeplitz matrices and R is also symmetric positive definite, is presented. The method exploits the fact that the trace of TR/sup -1/ depends only on the sum of the diagonals of R/sup -1/, and not on the whole matrix R/sup -1/. To obtain this sum, a fast efficient technique, built upon the Trench (1964) algorithm for computing the inverse of a Toeplitz matrix, is developed. The complexity of the algorithm depends on the generation function of matrix R and is O(N ln N) for generic functions and O(p ln p) for AR(p) functions.

This paper presents an effective algorithm for absolute phase (not simply modulo-2-pi) estimation from incomplete, noisy and modulo-2pi observations in interferometric aperture radar and sonar (InSAR/InSAS). The adopted framework is also representative of other applications such as optical interferometry, magnetic resonance imaging and diffraction tomography. The Bayesian viewpoint is adopted; the observation density is 2-pi-periodic and accounts for the interferometric pair decorrelation and system noise; the a priori probability of the absolute phase is modeled by a compound Gauss-Markov random field (CGMRF) tailored to piecewise smooth absolute phase images. We propose an iterative scheme for the computation of the maximum a posteriori probability (MAP) absolute phase estimate. Each iteration embodies a discrete optimization step (Z-step), implemented by network programming techniques and an iterative conditional modes (ICM) step (pi-step). Accordingly, the algorithm is termed ZpiM, where the letter M stands for maximization. An important contribution of the paper is the simultaneous implementation of phase unwrapping (inference of the 2pi-multiples) and smoothing (denoising of the observations). This improves considerably the accuracy of the absolute phase estimates compared to methods in which the data is low-pass filtered prior to unwrapping. A set of experimental results, comparing the proposed algorithm with alternative methods, illustrates the effectiveness of our approach.

The paper proposes a novel methodology to estimate the velocity of fast moving targets using a single synthetic aperture radar sensor without increasing the pulse repetition frequency. The basic reasoning is that, although the returned echoes may be aliased in the azimuth direction, its phase and amplitude are informative with respect to the moving target trajectory parameters. Based on this knowledge, we built an estimator for the moving targets' velocities and azimuthal positions. These parameters allow focusing of fast moving targets on their correct azimuth positions.

This paper presents a novel method to determine the complete velocity vector of a moving target using a single synthetic aperture radar (SAR) sensor. The method exploits the structure of the returned echo from a moving target: in the slow-time frequency domain, it is a scaled and shifted replica of the antenna radiation pattern immersed in Gaussian noise; the scale and the shift are related with the slant-range and the cross-range velocities, respectively. A Bayesian approach is then adopted to derive an estimator for the velocity vector. Simulation results illustrating the estimator effectiveness are presented.

This paper proposes a new nonparametric method for estimation of spectral moments of a zero-mean Gaussian process immersed in additive white Gaussian noise. Although the technique is valid for any order moment, particular attention is given to the mean Doppler (first moment) and to the spectral width (square root of the centered second spectral moment). By assuming that the power spectral density (PSD) of the underlying process is bandlimited, the maximum-likelihood estimates of its spectral moments are derived. A suboptimal estimate based on the sample covariance is also studied. Both methods are robust in the sense that they do not rely on any assumption concerning the PSD (besides being bandlimited). Under weak conditions, the set of estimates based on sample covariance is unbiased and strongly consistent. Compared with the classical pulse pair and the periodogram-based estimators, the proposed methods exhibit better statistical properties for asymmetric spectra and/or spectra with large spectral widths, while involving a computational burden of the same order.

This paper proposes a restoration scheme for noisy images generated by coherent imaging systems (e.g., synthetic aperture radar, synthetic aperture sonar ultrasound imaging, and laser imaging). The approach is Bayesian: the observed image intensity is assumed to be a random variable with gamma density; the image to be restored (mean amplitude) is modeled by a compound Gauss-Markov random field which enforces smoothness on homogeneous regions while preserving discontinuities between neighboring regions. A Neyman-Pearson detection criterion is used to infer the discontinuities, thus allowing to select a given false alarm probability maximizing the detection probability. The whole restoration scheme is then cast into a maximum a posteriori probability (MAP) problem. An expectation maximization type iterative scheme embedded in a continuation algorithm is used to compute the MAP solution. An application example performed on radar data is presented.

Presents a new method for endocardial (inner) and epicardial (outer) contour estimation from sequences of echocardiographic images. The framework herein introduced is fine-tuned for parasternal short axis views at the papillary muscle level. The underlying model is probabilistic; it captures the relevant features of the image generation physical mechanisms and of the heart morphology. Contour sequences are assumed to be two-dimensional noncausal first-order Markov random processes; each variable has a spatial index and a temporal index. The image pixels are modeled as Rayleigh distributed random variables with means depending on their positions (inside endocardium, between endocardium and pericardium, or outside pericardium). The complete probabilistic model is built under the Bayesian framework. As estimation criterion the maximum a posteriori (MAP) is adopted. To solve the optimization problem, one is led to (joint estimation of contours and distributions' parameters), the authors introduce an algorithm herein named iterative multigrid dynamic programming (IMDP). It is a fully data-driven scheme with no ad-hoc parameters. The method is implemented on an ordinary workstation, leading to computation times compatible with operational use. Experiments with simulated and real images are presented.